Table of contents. An “*" indicates nice. Number lists are here.

S000001 Number of primitive prime factors of 3^n - 1.

S000002 Number of primitive prime factors of 4^n - 1.

S000003 Number of primitive prime factors of 5^n - 1.

S000004 Number of primitive prime factors of 6^n - 1.

S000005 Number of primitive prime factors of 7^n - 1.

S000006 Number of primitive prime factors of 8^n - 1.

S000007 Number of primitive prime factors of 9^n - 1.

S000008 Number of primitive prime factors of 3^n + 1.

S000009 Number of primitive prime factors of 4^n + 1.

S000010 Number of primitive prime factors of 5^n + 1.

S000011 Number of primitive prime factors of 6^n + 1.

S000012 Number of primitive prime factors of 7^n + 1.

S000013 Number of primitive prime factors of 8^n + 1.

S000014 Number of primitive prime factors of 9^n + 1.

S000015 Number of primitive prime factors of 10^n + 1.

S000016 Numbers that are unique-period primes in bases 2 and 3.

S000017 Numbers that are unique-period primes in bases 2 and 4.

S000018 Numbers that are unique-period primes in bases 3 and 9.

S000019 Unique-period primes in base 3 in the order they are found.

S000020 Unique-period primes in base 4 in the order they are found.

S000021 Unique-period primes in base 5 in the order they are found.

S000022 Unique-period primes in base 6 in the order they are found.

S000023 Unique-period primes in base 7 in the order they are found.

S000024 Unique-period primes in base 8 in the order they are found.

S000025 Unique-period primes in base 9 in the order they are found.

S000026 Unique-period primes in base 3, sorted.

S000027 Unique-period primes in base 4, sorted.

S000028 Unique-period primes in base 5, sorted.

S000029 Unique-period primes in base 6, sorted.

S000030 Unique-period primes in base 7, sorted.

S000031 Unique-period primes in base 8, sorted.

S000032 Unique-period primes in base 9, sorted.

S000033 Unique-period primes in base 3, written in base 3.

S000034 Unique-period primes in base 4, written in base 4.

S000035 Unique-period primes in base 5, written in base 5.

S000036 Unique-period primes in base 6, written in base 6.

S000037 Unique-period primes in base 7, written in base 7.

S000038 Unique-period primes in base 8, written in base 8.

S000039 Unique-period primes in base 9, written in base 9.

S000040 Irregular table of primitive prime factors of 5^n - 1.

S000041 Irregular table of primitive prime factors of 6^n - 1.

S000042 Irregular table of primitive prime factors of 7^n - 1.

S000043 Irregular table of primitive prime factors of 8^n - 1.

S000044 Irregular table of primitive prime factors of 9^n - 1.

S000045 Irregular table of primitive prime factors of 3^n + 1.

S000046 Irregular table of primitive prime factors of 4^n + 1.

S000047 Irregular table of primitive prime factors of 5^n + 1.

S000048 Irregular table of primitive prime factors of 6^n + 1.

S000049 Irregular table of primitive prime factors of 7^n + 1.

S000050 Irregular table of primitive prime factors of 8^n + 1.

S000051 Irregular table of primitive prime factors of 9^n + 1.

S000052 Irregular table of primitive prime factors of 10^n + 1.

S000053 Number of 2's in the n X n multiplication table (mod n).

S000054 Composite numbers n that divide the sum of the composite numbers up to n.

S000055 Numbers n that divide the sum of the nonprime numbers up to n.

S000056 Numbers n that divide the sum of the composite numbers up to n.

S000057 Irregular table of conjectured Fibonacci numbers with exactly n bits set in binary.

S000058 Lucas numbers written in binary notation.

S000059 Irregular table of conjectured Lucas numbers with exactly n bits set in binary.

S000060 Conjectured number of Lucas numbers with exactly n bits set in their binary representation.

S000061* Symmetric n x n matrix of binomial coefficients.

S000062* Inverse of the symmetric n x n matrix of binomial coefficients in S000061.

S000063 Permanent of the symmetric n x n matrix of binomial coefficients in S000061.

S000064* Conjectured maximum number of primes between n*k and n*(k+1) for some k > 0.

S000065 Conjectured value of k for numbers in S000064.

S000066 Irregular triangle of the offsets used in S000064 and S000065.

S000067* First number of 3 consecutive twin prime pairs whose first numbers differ by 18 or less.

S000068* First number of 4 consecutive twin prime pairs whose first numbers differ by 30 or less.

S000069* First number of 5 consecutive twin prime pairs whose first numbers differ by 36 or less.

S000070* First number of 6 consecutive twin prime pairs whose first numbers differ by 48 or less.

S000071 Cases in a 1978 theorem of Ecklund, Eggleton, Erdos, and Selfridge.

S000072 Cases in a 1978 corollary of Ecklund, Eggleton, Erdos, and Selfridge.

S000073* Minimum length of an interval containing n twin primes, measured mid-to-mid.

S000074 Minimum number of numbers in an interval containing n twin primes.

S000075 Primes of the form x^2 + y^2 + z^2 + x*y + x*z + y*z for positive x, y, and z.

S000076 Primes not of the form x^2 + y^2 + z^2 + x*y + x*z + y*z for positive x, y, and z.

S000077 Primes of the form x^2 + y^2 + z^2 + x*y + x*z + y*z for nonnegative x, y, and z.

S000078 Primes not of the form x^2 + y^2 + z^2 + x*y + x*z + y*z for nonnegative x, y, and z.

S000079 Numbers that are the difference of two perfect numbers.

S000080 Primes p at which (nextprime(p) - p) / (p - prevprime(p)) is a new record high.

S000081 Primes p at which (nextprime(p) - p) / (p - prevprime(p)) is a new record low.

S000082 Primes p at which (nextprime(p) - p) / (p - prevprime(p)) is a new record high or a record low.

S000083 Cumulative sum of the squares of nonprimes.

S000084 Cumulative sum of the cubes of nonprimes.

S000085 Cumulative sum of the 4-th powers of nonprimes.

S000086 Cumulative sum of the 5-th powers of nonprimes.

S000087 Cumulative sum of the squares of composite numbers.

S000088 Cumulative sum of the cubes of composite numbers.

S000089 Cumulative sum of the 4-th powers of composite numbers.

S000090 Cumulative sum of the 5-th powers of composite numbers.

S000091* The matrix whose n-th row consists of increasing numbers whose largest prime factor is

prime(n).

S000092 Length of Lucas(n) when written in binary.

S000093 Numbers n such that x^2 + y^2 (mod n) assumes every number in Zn, where x and y are

nonzero.

S000094 Numbers n such that x^2 + y^2 (mod n) does not assume every number in Zn, where x and y

are nonzero.

S000095 Numbers n such that Fib(2n-1) and Fib(2n+1) are both prime.

S000096 Primes p such that p + 270 is also prime.

S000097 Table of initial primes such that there are k consecutive primes that are equal (mod n), with

k > 1 and n > 2.

S000098 Figurate primes, numbers of the form binomial(p^x,y), where p is prime and x and y are

positive integers.

S000099 Least k such that 10^n + k, 10^n + k + 2, and 10^n + k + 6 are prime.

S000100 Least k such that 10^n + k, 10^n + k + 4, and 10^n + k + 6 are prime.

S000101 Offset of the first prime triplet greater than 10^n; minimum of S000100(n) and S000101(n).

S000102 Numbers n such that Fib(n-1) and Fib(n+1) are both prime.

S000103 Numbers n such that Lucas(n-1) and Lucas(n+1) are both prime.

S000104* Minimum length of an interval containing n twin primes.

S000105 First number of 7 consecutive twin prime pairs whose first numbers differ by 62 or less.

S000106 Number of different admissible twin-prime tuples.

S000107 Primes not of form x^2 + 11*y^2.

S000108 Primes not of form x^2 + 13*y^2.

S000109 Primes not of form x^2 + 14*y^2.

S000110 Primes not of form x^2 + 15*y^2.

S000111 Primes not of form x^2 + 17*y^2.

S000112 Primes not of form x^2 + 18*y^2.

S000113 Primes not of form x^2 + 19*y^2.

S000114 Discriminant of odd primitive forms of regular ternary forms.

S000115 Discriminant of even primitive forms of regular ternary forms.

S000116* Discriminant of even and odd primitive forms of regular ternary forms, sorted.

S000117* Discriminants and 6-term forms of even and odd primitive forms of regular ternary forms, sorted.

S000118* Lists of four numbers (a,b,c,d) such that a*x^2 + b*y^2 + c*z^2 + d*u^2 represents all

nonnegative numbers.

S000119 The number of numbers i such that p = i (mod n) for some powerful number p and 0 <= i < n.

S000120 The number of numbers i such that p = i (mod n) for no powerful number p and 0 <= i < n.

S000121 Triangle of the least powerful number p such that p = i (mod n), or 0 if no such p exists.

S000122 Lists of four numbers (a,b,c,d) such that a*x^2 + b*y^2 + c*z^2 + d*u^2 represents all

nonnegative numbers, sorted.

S000123 Prefixes (first three terms) appearing in quadruples of S000118 and S000122.

S000124 Indices of Fibonacci numbers that do not have a factor of the form 4k+1.

S000125 Number of ways that n can be written as the sum of four positive nondecreasing squares and a prime.

S000126 Number of ways that n can be written as the sum of four nonnegative nondecreasing squares and a

prime.

S000127 Number of ways that n can be written as the sum of four positive squares and a prime.

S000128 Number of ways that n can be written as the sum of four nonnegative squares and a prime.

S000129 Number of ways that n can be written as the sum of three positive nondecreasing squares and a prime.

S000130 Number of ways that n can be written as the sum of three nonnegative nondecreasing squares and a

prime.

S000131 Number of ways that n can be written as the sum of three positive squares and a prime.

S000132 Number of ways that n can be written as the sum of three nonnegative squares and a prime.

S000133 Number of ways that n can be written as the sum of two positive nondecreasing squares and a prime.

S000134 Number of ways that n can be written as the sum of two nonnegative nondecreasing squares and a

prime.

S000135 Number of ways that n can be written as the sum of two positive squares and a prime.

S000136 Number of ways that n can be written as the sum of two nonnegative squares and a prime.

S000137 Table of triples (a,b,c) of quadratic forms having the number of discriminants given in A107628.

S000138* Table of 4-tuples (d,a,b,c) of quadratic forms having the number of discriminants d given in A107628.

S000139 Negative of discriminants of quadratic forms in S000137.

S000140* The numerator of sigma(n)/n for the superabundant numbers n.

S000141* The denominator of sigma(n)/n for the superabundant numbers n.

S000142 Powers such that the interval to the next power contains exactly one prime.

S000143 Powers such that the interval to the previous power contains exactly one prime.

S000144 Powers such that the interval to the next power contains exactly two primes.

S000145 Powers such that the interval to the previous power contains exactly two primes.

S000146 Powers such that the interval to the next power contains exactly three primes.

S000147 Powers such that the interval to the previous power contains exactly three primes.

S000148 Powers such that the interval to the next power contains exactly four primes.

S000149 Powers such that the interval to the previous power contains exactly four primes.

S000150 Powers such that the interval to the next power contains exactly five primes.

S000151 Powers such that the interval to the previous power contains exactly five primes.

S000152 Primes p such that p + 246 is also prime.

S000153* Sine: table in which the k-th row is round(sin(2*Pi*n/10^k)) for n = 0..10^k.

S000154 Cosine: table in which the k-th row is round(cos(2*Pi*n/10^k)) for n = 0..10^k.

S000155 Tangent: table in which the k-th row is round(tan((Pi/2)*n/10^k)) for n = 0..10^k - 1.

S000156 Cotangent: table in which the k-th row is round(cot((Pi/2)*n/10^k)) for n = 1..10^k.

S000157 Arcsine: table in which the k-th row is round(arcsin(n/10^k)) for n = -10^k..10^k.

S000158 Arccosine: table in which the k-th row is round(arccos(n/10^k)) for n = -10^k..10^k.

S000159 Nonnegative numbers n for which n^2 - n + 41 is not squarefree.

S000160 Nonnegative numbers n for which n^2 - n - 56 is not squarefree.

S000161 Nonnegative numbers n for which n^2 - n - 65 is not squarefree.

S000162 First differences of numbers n for which n^2 - n + 41 is not squarefree (S000159).

S000163 Number beginning n consecutive numbers that are not cube-free.

S000164 Number beginning n consecutive numbers that are not free of fourth powers.

S000165 The most common digit in the decimal form of 2^n; the least one if there are duplicates.

S000166 Number of times the digit S000165(n) appears in the decimal form of 2^n.

S000167 The digit in the longest run in the decimal form of 2^n; the least one if there are duplicates.

S000168 Longest run of the digit S000167(n) in the decimal form of 2^n.

S000169 The most common digit in the decimal form of 3^n; the least one if there are duplicates.

S000170 Number of times the digit S000169(n) appears in the decimal form of 3^n.

S000171 The digit in the longest run in the decimal form of 3^n; the least one if there are duplicates.

S000172 Longest run of the digit S000171(n) in the decimal form of 3^n.

S000173 The most common digit in the decimal form of 5^n; the least one if there are duplicates.

S000174 Number of times the digit S000173(n) appears in the decimal form of 5^n.

S000175 The digit in the longest run in the decimal form of 5^n; the least one if there are duplicates.

S000176 Longest run of the digit S000175(n) in the decimal form of 5^n.

S000177 The most common digit in the decimal form of 7^n; the least one if there are duplicates.

S000178 Number of times the digit S000177(n) appears in the decimal form of 7^n.

S000179 The digit in the longest run in the decimal form of 7^n; the least one if there are duplicates.

S000180 Longest run of the digit S000179(n) in the decimal form of 7^n.

S000181 Degrees of minimal polynomials that have a Salem number less than 1.3 as a zero.

S000182* Number of polynomials of degree S000181(n) that have a Salem number less than 1.3 as a zero.

S000183* Degrees of minimal polynomials having a Salem number under 1.3 as a root, sorted.

S000184 Coefficients of the minimal polynomial having a Salem number under 1.3 as a root, sorted.

S000185 Primitive Pythagorean triples (a, b, c^3) for prime c.

S000186 Primitive Pythagorean triples (a, b, c^3) for positive integer c.

S000187 Number of pairs of twin primes between e^n and e^(n+1) (where e = 2.718281828…).

S000188 Number of pairs twin primes between f^n and f^(n+1) (where f = e^2 = (2.718281828…)^2).

S000189 Numbers n such that e^n is next to a prime number.

S000190 Prime number s(n) such that (primepi(p) - primepi(s(n)) / (p - s(n)) is smaller for greater prime p.

S000191 The indices of the primes in S000190.

S000192 Floor of the n-th zero of the Bessel function J0.

S000193 Floor of the n-th zero of the Bessel function J1.

S000194 Floor of the n-th zero of the Bessel function J2.

S000195 Floor of the n-th zero of the Bessel function J3.

S000196 Floor of the n-th zero of the Bessel function J4.

S000197 Floor of the n-th zero of the Bessel function J5.

S000198 Floor of the n-th zero of the Bessel function Y0.

S000199 Floor of the n-th zero of the Bessel function Y1.

S000200 Floor of the n-th zero of the Bessel function Y2.

S000201 Floor of the n-th zero of the Bessel function Y3.

S000202 Floor of the n-th zero of the Bessel function Y4.

S000203 Floor of the n-th zero of the Bessel function Y5.

S000204 Floor of the n-th zero of the Airy function Ai.

S000205 The rounded n-th zero of the Airy function Ai.

S000206 Floor of the n-th zero of the Airy function Bi.

S000207 The rounded n-th zero of the Airy function Bi.

S000208* For p = prime(n), the total of the absolute value of the differences of the inverses of the

numbers 1..p-1 (mod p).

S000209 Maximum difference between primes less than 2^n.

S000210 Maximum difference between primes less than e^n.

S000211 Numbers n such that w(n) = sum_{k=0..n} binomial(n,k) binomial(n+k,k)/(2*k-1) is prime.

S000212 Primes w(n) for the n in S000211.

S000213 Sum of the reciprocals of Catalan numbers (A000108).

S000214 Alternating sum of the reciprocals of Catalan numbers (A000108).

S000215 Sum of the reciprocals of even-index Catalan numbers (A048990).

S000216 Sum of the reciprocals of odd-index Catalan numbers (A024492).

S000217 Index of Fibonacci numbers not containing the number 1.

S000218 Index of Fibonacci numbers not containing the number 2.

S000219 Index of Fibonacci numbers not containing the number 3.

S000220 Index of Fibonacci numbers not containing the number 4.

S000221 Index of Fibonacci numbers not containing the number 6.

S000222 Index of Fibonacci numbers not containing the number 7.

S000223 Index of Fibonacci numbers not containing the number 8.

S000224 Index of Fibonacci numbers not containing the number 9.

S000225 Index of Lucas numbers not containing the number 0.

S000226 Index of Lucas numbers not containing the number 1.

S000227 Index of Lucas numbers not containing the number 2.

S000228 Index of Lucas numbers not containing the number 3.

S000229 Index of Lucas numbers not containing the number 4.

S000230 Index of Lucas numbers not containing the number 5.

S000231 Index of Lucas numbers not containing the number 6.

S000232 Index of Lucas numbers not containing the number 7.

S000233 Index of Lucas numbers not containing the number 8.

S000234 Index of Lucas numbers not containing the number 9.

S000235 Number of composite numbers less than prime(n)^2.

S000236* Pi/4 - 1/2.

S000237 Pi/3 - sqrt(3)/4.

S000238 Pi/6 - sqrt(3)/4.

S000239 Three periodic orbits in the 3x-1 iteration.

S000240* Length of the 3x-1 iteration applied to odd numbers.

S000241 Where records occur in the 3x-1 iteration.

S000242 The record values of the length of the 3x-1 iteration.

S000243 Pi/5 - sqrt((5 + sqrt(5))/32).

S000244 Pi/7 - sqrt((1 + sin(Pi/14))/8).

S000245 Pi/8 - sqrt(2)/4.

S000246 Pi/9 - sqrt((1 - sin(Pi/18))/8).

S000247 Pi/10 - sqrt((5 - sqrt(5))/32).

S000248 Pi/11 - sqrt((1 - sin(3*Pi/22))/8).

S000249* Pi/12 - 1/4.

S000250 Floor of 1000000 times the area of one circular segment outside the unit n-gon.

S000251 Floor of 1000000 times the area of the n circular segments outside the unit n-gon.

S000252 Length of the reduced 3x-1 iteration applied to odd numbers.

S000253 Where records occur in the reduced 3x-1 iteration.

S000254 The record values of the length of the reduced 3x-1 iteration.

S000255 The number of primes up to n divides the n-th Fibonacci number.

S000256* Numbers n such that the n-th prime divides the n-th Fibonacci number.

S000257 The S000256(n)-th prime.

S000258 Dropping times of odd numbers in the 3x+1 problem (or the Collatz problem).

S000259 Dropping times of numbers 3+4*n in the 3x+1 problem (or the Collatz problem).

S000260* Irregular triangle of a conjectured periodicity in the Collatz (3x+1) iteration.

S000261 Dropping times in the 3x+1 (Collatz) iteration of 2^n-1.

S000262 Stopping times in the 3x+1 (Collatz) iteration of 2^n-1.

S000263 Irregular triangle of numbers x such that primepi(n * x) = n + x.

S000264 Number of numbers x such that primepi(n * x) = n + x.

S000265 Last number x such that primepi(n * x) = n + x.

S000266 Least k such that 2^n - k - 2 and 2^n - k are twin primes.

S000267 Least k such that 2^n - k and 2^n - k + 2 are twin primes.

S000268* Irregular triangle of numbers that have the same length Collatz (3x+1) iteration.

S000269* Dropping patterns of the 3x+1 (Collatz) iteration.

S000270 All numbers n that solve x^2 = 57 * 2^n + 117440512.

S000271 All numbers n that solve x^2 = 165 * 2^n + 26404.

S000272 The last number (or 0 if there is none) in row n of the irregular triangle A177789 (Collatz related).

S000273 The nonzero terms of S000272.

S000274 Records in the lengths of the Collatz (3x+1) iteration.

S000275 Least integer c such that binomial(c,n) has n prime factors greater than n.

S000276 Smallest number k > 0 such that the interval [k^2,(k+1)^2] contains n pairs of twin primes.

S000277 Numbers beginning 23 consecutive numbers whose squares sum to a square.

S000278 Numbers beginning 26 consecutive numbers whose squares sum to a square.

S000279 Numbers beginning 33 consecutive numbers whose squares sum to a square.

S000280 Numbers beginning 47 consecutive numbers whose squares sum to a square.

S000281 Numbers beginning 50 consecutive numbers whose squares sum to a square.

S000282 Numbers beginning 59 consecutive numbers whose squares sum to a square.

S000283 Numbers n (excluding squares > 1) such that sum of squares of n consecutive integers >= 1 is a square.

S000284 Numbers beginning 96 consecutive numbers whose squares sum to a square.

S000285 Numbers beginning 338 consecutive numbers whose squares sum to a square.

S000286 Position of first zero digit in 2^n, counting from the right.

S000287 Position of first zero digit in 3^n, counting from the right.

S000288 Position of first zero digit in 4^n, counting from the right.

S000289 Position of first zero digit in 5^n, counting from the right.

S000290 Position of first zero digit in 6^n, counting from the right.

S000291 Position of first zero digit in 7^n, counting from the right.

S000292 Position of first zero digit in 8^n, counting from the right.

S000293 Position of first zero digit in 9^n, counting from the right.

S000294 Irregular triangle of numbers n such that 2^n does not contain 0, 1, …, 9.

S000295 Irregular triangle of numbers n such that 3^n does not contain 0, 1, …, 9.

S000296 Irregular triangle of numbers n such that 4^n does not contain 0, 1, …, 9.

S000297 Irregular triangle of numbers n such that 5^n does not contain 0, 1, …, 9.

S000298 Irregular triangle of numbers n such that 6^n does not contain 0, 1, …, 9.

S000299 Irregular triangle of numbers n such that 7^n does not contain 0, 1, …, 9.

S000300 Irregular triangle of numbers n such that 8^n does not contain 0, 1, …, 9.

S000301 Irregular triangle of numbers n such that 9^n does not contain 0, 1, …, 9.

S000303 Sequence S000302 without the zeros.

S000304 A chain of twin primes.

S000305* Another chain of twin primes.

S000306 Ratio (1+b(n+1)) / b(n) for the sequence b = S000304.

S000307 Factors used in forming the lower twin primes in S000305.

S000308 First prime of 9 primes in a range of 31 numbers.

S000309 First prime of 7 primes in a range of 21 numbers.

S000310 Least non-border binomial coefficient having n prime factors, or 0 if none exists.

S000311 Least row in the table of binomial coefficients having a number with n distinct prime factors.

S000312 For the n-th twin primes x±1, the least number k*x-1 such that k*x±1 are twin primes for k > 1.

S000313 For the n-th twin primes x±1, the least number k > 1 such that k*x±1 are twin primes.

S000314 Prime values of m^2 - 2*n^2 for prime m and n.

S000315 Numbers n such that 2n+1 and 2n+3 are prime.

S000316 Numbers n such that 2n+1, 2n+3, and 2n+7 are prime.

S000317 Numbers n such that 2n+1, 2n+3, 2n+7, and 2n+9 are prime.

S000318 Numbers n such that 2n+1, 2n+3, 2n+7, 2n+9, and 2n+13 are prime.

S000319 Numbers n such that 2n+5, 2n+7, and 2n+11 are prime.

S000320 Numbers n such that 2n+5, 2n+7, 2n+11, and 2n+13 are prime.

S000321 Numbers n such that 2n+5, 2n+7, 2n+11, 2n+13, and 2n+17 are prime.

S000322* Pairs of prime numbers {p1, p2} that list the range that can generate primes, as in S000321.

S000323 The index of Lucas numbers that are not squarefree.

S000324 Lucas numbers that are not squarefree.

S000325 The squarefull part of the n-th Lucas number.

S000326 The squarefull part of the n-th Fibonacci number.

S000327 Primes having numbers with 4 and 2 prime factors (counting multiplicity) before and after them.

S000328 Primes having numbers with 2 and 4 prime factors (counting multiplicity) before and after them.

S000329 Primes having numbers with 2 and 4 prime factors (counting multiplicity) before or after them.

S000330 Primes having numbers with a total of 6 prime factors (counting multiplicity) before or after them.

S000331 Records of the total number of prime factors (counted multiply) in the numbers between consecutive primes.

S000332 The first number in the pair shown in S000331.

S000333 The second number in the pair shown in S000331.

S000334 Records of the total number of prime factors in the numbers between consecutive primes.

S000335 The first number in the pair shown in S000334.

S000336 The second number in the pair shown in S000334.

S000337* Hypotenuses of right triangles having legs that are triangular numbers.

S000338* Three sides of right triangles having legs that are triangular numbers, ordered by hypotenuse.

S000339 Reduced hypotenuses of right triangles in which the hypotenuse and one leg are triangular numbers.

S000340* Three sides of right triangles having the hypotenuse and one leg that are triangular numbers.

S000341 Number of solutions to 4/prime(n) = 1/x + 1/y + 1/z for integers x, y, z with 0 < x < y < z.

S000342 Primes at which maximums occur in the Erdos-Straus conjecture graph (see S000341).

S000343 The maximums of the Erdos-Straus conjecture graph at the primes in S000342.

S000344 Numbers n such that the n-th triangular number n*(n+1)/2 is a palindrome in base 2.

S000345 Numbers n such that the n-th triangular number n*(n+1)/2 is a palindrome in base 3.

S000346 Numbers n such that the n-th triangular number n*(n+1)/2 is a palindrome in base 4.

S000347 Numbers n such that the n-th triangular number n*(n+1)/2 is a palindrome in base 5.

S000348 Numbers n such that the n-th triangular number n*(n+1)/2 is a palindrome in base 6.

S000349 Numbers n such that the n-th triangular number n*(n+1)/2 is a palindrome in base 7.

S000350 Numbers n such that the n-th triangular number n*(n+1)/2 is a palindrome in base 8.

S000351 Numbers n such that the n-th triangular number n*(n+1)/2 is a palindrome in base 9.

S000352 Palindromic triangular numbers in base 2.

S000353 Palindromic triangular numbers in base 3.

S000354 Palindromic triangular numbers in base 4.

S000355 Palindromic triangular numbers in base 5.

S000356 Palindromic triangular numbers in base 6.

S000357 Palindromic triangular numbers in base 7.

S000358 Palindromic triangular numbers in base 8.

S000359 Palindromic triangular numbers in base 9.

S000360 Numbers n such that 3^n-1 has only one primitive prime factor.

S000361 Numbers n such that 4^n-1 has only one primitive prime factor.

S000362 Numbers n such that 5^n-1 has only one primitive prime factor.

S000363 Numbers n such that 6^n-1 has only one primitive prime factor.

S000364 Numbers n such that 7^n-1 has only one primitive prime factor.

S000365 Numbers n such that 8^n-1 has only one primitive prime factor.

S000366 Numbers n such that 9^n-1 has only one primitive prime factor.

S000367 Numbers n such that 2^n+1 has only one primitive prime factor.

S000368 Numbers n such that 3^n+1 has only one primitive prime factor.

S000369 Numbers n such that 4^n+1 has only one primitive prime factor.

S000370 Numbers n such that 5^n+1 has only one primitive prime factor.

S000371 Numbers n such that 6^n+1 has only one primitive prime factor.

S000372 Numbers n such that 7^n+1 has only one primitive prime factor.

S000373 Numbers n such that 8^n+1 has only one primitive prime factor.

S000374 Numbers n such that 9^n+1 has only one primitive prime factor.

S000375 Numbers n such that 10^n+1 has only one primitive prime factor.

S000376 Irregular array of positive numbers that form palindromic tetrahedral numbers in bases 2 to 9.

S000377 Irregular array of positive palindromic tetrahedral numbers in bases 2 to 9.

S000378 (3^(4*n - 1) - 1)/2.

S000379 For the n in S000378, the tetrahedral number n*(n+1)*(n+2)/6 in base 9.

S000380 Irregular array of positive numbers that form palindromic square pyramidal numbers in bases 2 to 10.

S000381 Irregular array of positive palindromic square pyramidal numbers in bases 2 to 10.

S000382 Deteminant of the n x n symmetric matrix whose lower triangular part is the binomial numbers mod 2.

S000383 Let a(n) = 2^n + a(n-2) for n > 1, a(0) = 3, and a(1) = 3.

S000384 Numbers n such that the number of composite numbers around n^2 is a new record.

S000385 The record number of composite numbers around S000384(n)^2.

S000386 Number of solutions to 4/prime(n) = 1/x + 1/y + 1/z for integers x, y, z with 0 < x <= y <= z.

S000387 The practical part of positive integer n.

S000388 Numbers n for which a sum of n-th roots of unity is a new nonzero minimum (in absolute value).

S000389 Gaussian integers near the curve z = exp(I * t).

S000390 Gaussian primes near the curve z = exp(I * t).

S000391* Numbers n such that Gaussian primes x + I*n appear to have distributions similar to 4m+3 primes.

S000392 An adjacency matrix for the 10-node Petersen graph.

S000393 Eigenvalues of the adjacency matrix for the 10-node Petersen graph.

S000394 Palindromic primes in base 17, but written here in base 10.

S000395 Palindromic primes in base 18, but written here in base 10.

S000396 Palindromic primes in base 19, but written here in base 10.

S000397 Palindromic primes in base 20, but written here in base 10.

S000398 Palindromic primes in base 21, but written here in base 10.

S000399 Palindromic primes in base 22, but written here in base 10.

S000400 Palindromic primes in base 23, but written here in base 10.

S000401 Palindromic primes in base 24, but written here in base 10.

S000402 Palindromic primes in base 25, but written here in base 10.

S000403 The n-th palindromic prime in base n, but written here in base 10.

S000404 Triangle of the number of prime factors (not counted multiply) of first-quadrant Guassian integers.

S000405 Triangle of the number of prime factors (counted multiply) of first-quadrant Guassian integers.

S000406 Triangle of the number of divisors of first-quadrant Guassian integers.

S000407 The first Gaussian integer having n distinct prime factors in the triangle S000404.

S000408 Diagonal having the first Gaussian integer with n distinct prime factors in the triangle S000404.

S000409 Gaussian primes in the first quadrant, sorted by magnitude and real part.

S000410 Gaussian primes strictly in the first quadrant, sorted by magnitude and real part.

S000411 Gaussian primes strictly in the first quadrant, sorted by magnitude and real part < imaginary part.

S000412 Gaussian primes strictly in the first quadrant, sorted by magnitude and imaginary part < real part.

S000413 Gaussian primes strictly in the first quadrant, sorted by magnitude and imaginary part.

S000414 Gaussian primes in the first quadrant, sorted by magnitude and imaginary part.

S000415* The Gaussian primes, sorted by magnitude and angle.

S000416 Distance squared to the center of a 4-lion from the origin.

S000417 Position of the center of a 4-lion in the first quadrant, sorted by magnitude and real part.

S000418 Position of the center of a 4-lion in the octant x >= y >= 0.

S000419 Position of the center of a 4-lion in the octant y >= x >= 0.

S000420 Position of the center of a 4-lion in the first quadrant, sorted by magnitude and imaginary part.

S000421 Distance squared to the center of a “real" 4-lion from the origin.

S000422 Position of the center of a “real” 4-lion in the first quadrant, sorted by magnitude and real part.

S000423 Position of the center of a “real" 4-lion in the octant x >= y >= 0.

S000424 Position of the center of a “real" 4-lion in the octant y >= x >= 0.

S000425 Position of the center of a “real" 4-lion in the first quadrant, sorted by magnitude and imaginary part.

S000426* Sum of the absolute values of the coefficients of Phi(n,x), the n-th cyclotomic polynomial.

S000427* Sum of the squares of the coefficients of Phi(n,x), the n-th cyclotomic polynomial.

S000428* The Bombieri norm of Phi(n,x), the n-th cyclotomic polynomial.

S000429 The triangle of nonzero (m - 1) * (n - 1) for 1 < m < n.

S000430 The triangle of nonzero (m - 1) * (n - 1) - 1 for 1 < m < n.

S000431 Three-dimensional triangle of the conductor of a, b and c for the expression a*x + b*y + c*z.

S000432 Three-dimensional triangle of one less than the conductor of a, b, c for the expression a*x + b*y + c*z.

S000433 Conductor of prime(n), prime(n+1), and prime(n+2).

S000434 Conjectured value of the conductor of prime(n), prime(n+1), prime(n+2),….

S000435* Conjectued number of terms required for the conductor function in S000434 to converge.

S000436 Primes p such that the iteration in S000334 converges after more than primepi(p) steps.

S000437 Primes p such that the iteration in S000334 converges at or before primepi(p) steps.

S000438* Triangle of the power tower k^k^k^... (mod n) for k = 1..(n-1).

S000439 The height of the exponents in S000438.

S000440 Number of odd primes in Collatz sequences having only 1, even numbers, and primes.

S000441* The first prime that produces exactly n odd primes in the Collatz iteration; the other terms are 1 or even.

S000442 Number of primes in Collatz sequences having only 1, even numbers, and primes.

S000443 Prime numbers whose Collatz sequence contains only 3 primes.

S000444 Odd numbers n such that evenN(c)/oddN(c) is a new record, where c is the Collatz iteration of n.

S000445 Numbers n such that (5*2^n - 1)/3 is prime.

S000446 Numbers n for which there are no base-2 Fermat pseudoprimes x that have ord(2,x) = n.

S000447* Irregular table of k! mod p for k = 1..p-1 for primes p.

S000448 Number of distinct values among the p-1 terms k! mod p for k = 1..p-1 for primes p.

S000449 Number of nondistinct values among the p-1 terms k! mod p for k = 1..p-1 for primes p.

S000450 Numbers not in the irregular table of k! mod p for k = 1..p-1 for primes p.

S000451 Least number not in the irregular table of k! mod p for k = 1..p-1 for primes p.

S000452 p - 1 - greatest number not in the irregular table of k! mod p for k = 1..p-1 for primes p.

S000453 Points higher than earlier points in the graph of A229037.

S000454 The x-coordinates of the points in S000453.

S000455 The y-coordinates of the points in S000453.

S000456 Number of ways in which 2n-1 can be written as the sum of 3 odd composite numbers 7 < u < v < w.

S000457 Number of ways in which 2n-1 can be written as the sum of 3 odd composite numbers 9 <= u <= v <= w.

S000458 Number of ways prime(n) is a sum of 3 odd nonprimes r,s,t satisfying 9 <= r <= s <= t.

S000459 Number of ways in which 2n-1 can be written as the sum of 3 odd composite numbers.

S000460 Number of ways in which prime(n) can be written as the sum of 3 odd composite numbers.

S000461 Let s = {1,2,…,n}. Then a(n) = total(d(s))^2 - total(d(s)^3), where d(s) is the number of divisors.

S000462 Same as S000461 except that the set s is a subset of {1,2,…,n}.

S000463 Number of terms in S000461(n) that are zero.

S000464 Sequence S000462 with each row sorted.

S000465 Numbers n such that max(sfp(n), sfp(n+1), sfp(n+2)) = n, where spf(n) is the squarefree part of n.

S000466 Numbers n such that max(sfp(n), sfp(n+1), sfp(n+2)) < sqrt(n), where spf(n) is the squarefree part of n.

S000467* Numbers n such that max(sfp(n), sfp(n+1), sfp(n+2)) < n^(1/3), where spf(n) is the squarefree part of n.

S000468 Alternating sum of the number of distinct prime factors of the numbers up to n.

S000469 Alternating sum of the number of prime factors (counted multiply) of the numbers up to n.

S000470 Alternating sum of the number of distinct prime factors of the numbers up to 2^n.

S000471 Alternating sum of the number of prime factors (counted multiply) of the numbers up to 2^n.

S000472 Primes p producing the longest possible period in the Fibonacci 3-step (mod p) sequence.

S000473 Primes p producing the longest possible period in the Fibonacci 4-step (mod p) sequence.

S000474 Primes p producing the longest possible period in the Fibonacci 5-step (mod p) sequence.

S000475 Primes p such that the polynomial x^4-x^3-x^2-x-1 mod p has 1 distinct zero.

S000476 Primes p such that the polynomial x^4-x^3-x^2-x-1 mod p has 2 distinct zeros.

S000477 Primes p such that the polynomial x^5-x^4-x^3-x^2-x-1 mod p has 1 distinct zero.

S000478 Primes p such that the polynomial x^5-x^4-x^3-x^2-x-1 mod p has 2 distinct zeros.

S000479 Primes p such that the polynomial x^5-x^4-x^3-x^2-x-1 mod p has 3 distinct zeros.

S000480 Record minimum values in A092693 = sum of iterated phi(n).

S000481* Squares that are the sum of 4 unequal squares of Fibonacci numbers.

S000482 Square root of the numbers in S000481.

S000483 Unique numbers in S000481.

S000484 Square root of the numbers in S000483.

S000485 Five-tuples of squares used in S000481.

S000486 Square root of the five-tuples of squares used in S000481.

S000487 Squares that are the sum of 4 different squares of Fibonacci numbers.

S000488 Square root of numbers in S000487.

S000489 Squares that are the sum of 4 different squares of Fibonacci numbers, with 1 being the least.

S000490 Square root of S000489.

S000491 Squares that are the sum of 4 unequal squares of Lucas numbers.

S000492 Square root of the numbers in S000491.

S000493 Five-tuples of squares used in S000491.

S000494 Square root of the five-tuples used in S000491.

S000495* Squares that are the sum of 5 unequal squares of Lucas numbers.

S000496 Square root of the numbers in S000495.

S000497 Square root of numbers that are the sum of 5 unequal squares of Lucas numbers, case 1.

S000498 Square root of numbers that are the sum of 5 unequal squares of Lucas numbers, case 2.

S000499 Square root of numbers that are the sum of 5 unequal squares of Lucas numbers, case 3.

S000500 Square root of numbers that are the sum of 5 unequal squares of Lucas numbers, union.

S000501* Primitive Pythagorean triples in which the two legs are palindromes.

S000502 Primitive Pythagorean triples (in reverse order) in which the two legs are palindromes.

S000503* Primitive Pythagorean triples in which the hypotenuse and a leg are palindromes.

S000504 Primitive Pythagorean triples (in reverse order) in which the hypotenuse and a leg are palindromes.

S000505* Primitive Pythagorean triples in which two parts are palindromes.

S000506 Primitive Pythagorean triples (in reverse order) in which two parts are palindromes.

S000507 The hypotenuse of a primitive Pythagorean triangle in which the two legs are palindromes.

S000508 The hypotenuse of a primitive Pythagorean triangle in which the hypotenuse and a leg are palindromes.

S000509 The hypotenuse of primitive Pythagorean triples in which two parts are palindromes.

S000510 Primitive Pythagorean triples in which the hypotenuse and a leg are palindromes in base 2.

S000511 Primitive Pythagorean triples (in reverse order) in which the hypotenuse and a leg are palindromes in base 2.

S000512 The hypotenuse of a primitive Pythagorean triangle in which the hypotenuse and a leg are palindromes in base 2.

S000513 Numerators of a Farey series.

S000514 Denominators of a Farey series.

S000515 The index of the terms in a Farey sequence that are integers.

S000516 The value of the terms in a Farey sequence that are integers.

S000517 Pairs giving the number of primes less than 10^n of the forms 4k+1 and 4k+3.

S000518 Triples giving the number of primes less than 10^n of the forms 6k+1, 6k+3, and 6k+5.

S000519 Quadruples giving the number of primes less than 10^n of the forms 8k+1, 8k+3,…, 8k+7.

S000520 5-tuples giving the number of primes less than 10^n of the forms 10k+1, 10k+3,…, 10k+9.

S000521 6-tuples giving the number of primes less than 10^n of the forms 12k+1, 12k+3,…, 12k+11.

S000522 7-tuples giving the number of primes less than 10^n of the forms 14k+1, 14k+3,…, 14k+13.

S000523 8-tuples giving the number of primes less than 10^n of the forms 16k+1, 16k+3,…, 16k+15.

S000524 16-tuples giving the number of primes less than 10^n of the forms 32k+1, 32k+3,…, 32k+31.

S000525 32-tuples giving the number of primes less than 10^n of the forms 64k+1, 64k+3,…, 64k+63.

S000526* Firoozbakht conjecture related: ceiling(prime(n)^(1+1/n)) - prime(n+1).

S000527 Conjectured maximum value k such that n = ceiling(prime(k)^(1+1/k)) - prime(k+1), or 0 if none such.

S000528 Conjectured maximum prime p such that n = ceiling(p^(1+1/k)) - nextprime(p), or 0 if none such.

S000529 Number of composite numbers around 2^n.

S000530 Number of composite numbers around 3^n.

S000531 Number of composite numbers around 4^n.

S000532 Number of composite numbers around 5^n.

S000533 Number of composite numbers around 6^n.

S000534 Number of composite numbers around 7^n.

S000535 Number of composite numbers around 8^n.

S000536 Number of composite numbers around 9^n.

S000537 Conjectured sorted differences between adjacent powers.

S000538 Position of S000537(n) in the list of powers A001597.

S000539* Numbers n such that the distance between Riemann zeros r(n+1) - r(n) is a new minimum.

S000540 Last i for which the difference of the Riemann zeros r(i+1) - r(i) > (7 - n)*I.

S000541* Irregular triangle of adjacency matrices of simple connected graphs on n points.

S000542 First i for which the difference of the Riemann zeros r(i+1) - r(i) < 2^(-n) * I.

S000543 Number of composite numbers around n!.

S000544 Number of composite numbers around n^n.

S000545 Number of primes in the range n^2 to (n + log(n)/2)^2.

S000546 Number of primes in the range n^2 to (n + 1/log(n))^2.

S000547 Number of primes in the interval p to p + floor(log(p)^2), where p is prime(n).

S000548 Least prime p such that the interval p to p + floor(log(p)^2) contains exactly n primes.

S000549 Number of primitive roots that are minimal.

S000550 The last number having S000549(n) primitive roots.

S000551 Irregular triangle in which row n has the numbers having S000549(n) primitive roots.

S000552 Numbers that occur as the number of primitive roots.

S000553 Sum of the primitive roots of n (mod n).

S000554 The generalized Collatz iteration, n*X+1, is finite for these n, starting from X=3.

S000555 Primes p such that p+2 is prime and p+6 has 2 or fewer prime factors.

S000556 Primes p such that p+2 is prime and p+6 has 3 or fewer prime factors.

S000557 Primes p such that p+2 and p+6 have 2 or fewer prime factors.

S000558 Primes p such that p+2 has 2 or fewer prime factors and p+6 has 3 or fewer prime factors.

S000559 Record pairs (n,k) of odd numbers n that have a record 3x+1 (Collatz) length k.

S000560 The numbers n in S000559.

S000561 The numbers k in S000559.

S000562 The least prime number p such that p+2 is prime and p+6 has n prime factors (counted multiply).

S000563 The least prime number p such that p+2 has up to two prime factors and p+6 has n prime factors.

S000564 Number of composite numbers between S000564(n) and the next prime number.

S000565 Number of composite numbers between S000564(n) and the next prime number.

S000566* Smallest prime of the form xxxx…xxx n for odd n and digit x, or 0 if there is none.

S000567* Smallest prime of the form n xxxx…xxx for digit x, or 0 if there is none.

S000568 First prime that begins a range of composite numbers containing n squarefree numbers.

S000569 Number of composite numbers in the range implied by S000568(n).

S000570 First prime that begins a range of composite numbers containing n powers of numbers.

S000571 Composite numbers (p*q*r*…)^k where p, q, r,… are distinct primes and k is any positive integer.

S000572 Triangle of bases for which the least number is palindromic for n bases.

S000573* Irregular rows of bases for which n is palindromic.

S000574 Primes p for which there is more than one base in which p is palindromic.

S000575 Irregular array of primitive weird numbers of the form 2^n p*q.

S000576 Number of primitive weird numbers of the form 2^n p*q.

S000577* Least prime p such that the interval p to p + floor(log(p)^2) contains at least n primes.

S000578 Sequence S000577 without duplicates.

S000579 Primitive Pythagorean triples in which the two legs and the hypotenuse are palindromes in base 9, shown in base 10.

S000580* Primitive Pythagorean triples in which the two legs and the hypotenuse are palindromes in base 9, shown in base 9.

S000581 Number of times n can be written as a square or the sum of a square and a prime.

S000582 The numbers n where records occur in S000581.

S000583 Nonsquare numbers n for which S000581(n) = 1.

S000584 Least k > 1 so that k^n + (k-1)^n >= (k+1)^n + 1^n.

S000585 Pairs of positive numbers x <= y such that x^2 + y^2 is prime.

S000586 Pairs of positive numbers x <= y such that x^4 + y^4 is prime.

S000587 Pairs of positive numbers x <= y such that x^8 + y^8 is prime.

S000588 Pairs of positive numbers x <= y such that x^16 + y^16 is prime.

S000589 Pairs of positive numbers x <= y such that x^32 + y^32 is prime.

S000590 Pairs of positive numbers x <= y such that x^64 + y^64 is prime.

S000591 Pairs of positive numbers x <= y such that x^2 + y^2 is prime and increasing.

S000592 Pairs of positive numbers x <= y such that x^4 + y^4 is prime and increasing.

S000593 Pairs of positive numbers x <= y such that x^8 + y^8 is prime and increasing.

S000594 Pairs of positive numbers x <= y such that x^16 + y^16 is prime and increasing.

S000595 Pairs of positive numbers x <= y such that x^32 + y^32 is prime and increasing.

S000596 Pairs of positive numbers x <= y such that x^64 + y^64 is prime and increasing.

S000597 Number of pairs of numbers (n,x) with 0 < x <= n such that n^2 + x^2 is prime.

S000598 Number of pairs of numbers (n,x) with 0 < x <= n such that n^4 + x^4 is prime.

S000599 Number of pairs of numbers (n,x) with 0 < x <= n such that n^8 + x^8 is prime.

S000600 Number of pairs of numbers (n,x) with 0 < x <= n such that n^16 + x^16 is prime.

S000601 Number of pairs of numbers (n,x) with 0 < x <= n such that n^32 + x^32 is prime.

S000602 Number of pairs of numbers (n,x) with 0 < x <= n such that n^64 + x^64 is prime.

S000603* Prime numbers that are “popular”.

S000604 The number at which a prime in S000603 first becomes “popular”.

S000605 Irregular table of numbers beginning prime(n)^2 consecutive numbers whose squares sum to square.

S000606 The number of terms in row n of S000605.

S000607 Primes that satisfy p(k+1) - p(k) > p(k+3) - p(k+1), where p(k) is the k-th prime.

S000608 Irregular table of numbers k such that the sum of the squares of n to n+k is a square.

S000609 Irregular table of numbers k such that the sum of the squares of k to n is a square.

S000610 First number having n representations as the sum of up to 9 cubes.

S000611 Conjectured number of numbers requiring exactly n cubes to represent.

S000612* Conjectured largest number requiring exactly n cubes to represent.

S000613 The product of the composite numbers in the first prime gap having 2*n - 1 numbers.

S000614 Next prime after A000230(n).

S000615 A stable set of primes of the form j^2 + k^2 with all j and k distinct.

S000616 Pairs of numbers (j,k) that produce the primes in S000615.

S000617 A stable set of primes of the form j^4 + k^4 with all j and k distinct.

S000618 Pairs of numbers (j,k) that produce the primes in S000617.

S000619 Number of ways to pair up {1^4, 2^4, ..., (2n)^4 } so the sum of each of the n pairs is prime.

S000620 Irregular table of m! (mod p), where p = prime(n) and m varies from 1 to p.

S000621 Number of numbers <= p that are not among the values m! (mod p), where p = prime(n) and m <= p.

S000622 Primes that produce decreasing values of S000621(k)/prime(k).

S000623 Zero followed by Fibonacci(2*n+1) for n = 1, 2, 3,....

S000624 Least number k such that k^2 + n^2 is a square, or zero if no such square exists.

S000625 Least number k such that k^2 + n^2 is a square and gcd(k,n) = 1, or zero if no such square exists.

S000626 Iterates of the Riesel problem starting at 509203.

S000627* Numbers that are sums of consecutive squares but not squares.

S000628 Number of numbers between (n-1)^2 and n^2 that are the sum of consecutive squares.

S000629* Numbers that are sums of consecutive cubes but not cubes.

S000630 Number of numbers between (n-1)^3 and n^3 that are the sum of consecutive cubes.

S000631* Numbers that are sums of consecutive fourth powers but not fourth powers.

S000632 Number of numbers between (n-1)^4 and n^4 that are the sum of consecutive fourth powers.

S000633* Numbers that are sums of consecutive fifth powers but not fifth powers.

S000634 Number of numbers between (n-1)^5 and n^5 that are the sum of consecutive fifth powers.

S000635* Numbers that are sums of consecutive sixth powers but not sixth powers.

S000636 Number of numbers between (n-1)^6 and n^6 that are the sum of consecutive sixth powers.

S000637 The number of ways number n can be written as the sum of a squarefree number and a prime squared.

S000638 Numbers that cannot be written as the sum of a squarefree number and a prime squared.

S000639 Least prime p such that there are n numbers m such that p divides m! + 1.

S000640 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x+3478727)^2 = y^2.

S000641 Sequential product of primes of the form 8k ± 1.

S000642* Order of the linear difference equation that solves the Diophantine equation x^2 + (x+n)^2 = y^2.

S000643* Irregular table of difference equations and initial terms descibed in S000642.

S000644 Least lower twin prime of the form 2*n*k ± 1.

S000645 Lower prime of the twin prime pair of the form 16*k ± 1.

S000646 Lower prime of the twin prime pair of the form 24*k ± 1.

S000647 Sorted numbers that are the product of primes of the form 8k ± 1 with non-increasing exponents.

S000648 The order of the difference equation that solves the equation x^2 + (x + S000647(n))^2 = y^2.

S000649 Least k for which the linear difference equation that solves x^2 + (x+k)^2 = y^2 has order 4n-1.

S000650 Largest prime factor of 3^n - 2^n.

S000651 Largest prime factor of 4^n - 3^n.

S000652 Largest prime factor of 5^n - 2^n.

S000653 Largest prime factor of 5^n - 3^n.

S000654 Largest prime factor of 5^n - 4^n.

S000655 Largest prime factor of 6^n - 5^n.

S000656 Largest prime factor of 7^n - 2^n.

S000657 Largest prime factor of 7^n - 3^n.

S000658 Largest prime factor of 7^n - 4^n.

S000659 Largest prime factor of 7^n - 5^n.

S000660 Largest prime factor of 7^n - 6^n.

S000661 Largest prime factor of 8^n - 3^n.

S000662 Largest prime factor of 8^n - 5^n.

S000663 Largest prime factor of 8^n - 7^n.

S000664 Largest prime factor of 9^n - 2^n.

S000665 Largest prime factor of 9^n - 4^n.

S000666 Largest prime factor of 9^n - 5^n.

S000667 Largest prime factor of 9^n - 7^n.

S000668 Largest prime factor of 9^n - 8^n.

S000669 Numbers n such that only one number m exists such that n divides m!+1.

S000670 Primes p such that there are 2 values of m such that p divides m! + 1.

S000671 Primes p such that there are 3 values of m such that p divides m! + 1.

S000672 Primes p such that there are 4 values of m such that p divides m! + 1.

S000673 Primes p such that there are 5 values of m such that p divides m! + 1.

S000674 Primes p such that there are 6 values of m such that p divides m! + 1.

S000675 Primes p such that there are 7 values of m such that p divides m! + 1.

S000676 The composite number A256519(n) divides a(n)! + 1.

S000677 Pairs of numbers A256519(n) and S000676(n).

S000678 The n-th row has the numbers r such that prime(n) divides r! + 1.

S000679 Number of distinct primes among the squares mod prime(n).

S000680 The n-th row has the n primes in the interval 100*k(n) to 100*k(n)+99, where k(n) is in A186311.

S000681* An encoding of prime constellations.

S000682 Least prime that begins the constellation given by S000681(n).

S000683 Irregular triangle whose n-th row gives the constellation encoded in S000681(n).

S000684 Triangle whose n-th row has n numbers m such that prime S000639(n) divides m! + 1.

S000685 The n+1 degree polynomial which computes A064538(n) times the sum of the n-th powers of positive integers (constant term first).

S000686 The n+1 degree polynomial which computes A064538(n) times the sum of the n-th powers of positive integers (constant term last).

S000687* Number of times that the sum of 4 unordered triangular numbers equals n.

S000688 Position of first occurrence of n different digits in the number E.

S000689 Position of first occurrence of n different digits in the number Pi.

S000690 Position of the last occurrence of n digits in the number Pi that do not occur earlier.

S000691 Number of integer points in the 3-dimensional sliver in the n x n x n cube.

S000692 Number of integer points in the 4-dimensional sliver in the n x n x n x n 4-cube.

S000693 Number of integer points in the 5-dimensional sliver in the n x n x n x n x n 5-cube.

S000694 Number of divisors of n^2 - 1.

S000695 Sum of the number of divisors of i^2 - 1 for i = 2..n.

S000696 Numbers n at which the number of divisors of n^2 -1 reaches a new record.

S000697 For the numbers n in S000696, the number of divisors of n^2 - 1.

S000698* Number of palindromic triples that sum to n.

S000699 Number of palindromic pairs that sum to n.

S000700 Number of palindromic (in base 9) triples that sum to n.

S000701 Number of palindromic (in base 8) triples that sum to n.

S000702 Number of palindromic (in base 7) triples that sum to n.

S000703 Number of palindromic (in base 6) triples that sum to n.

S000704 Number of palindromic (in base 5) triples that sum to n.

S000705 Number of palindromic (in base 4) triples that sum to n.

S000706 Number of palindromic (in base 3) triples that sum to n.

S000707 Number of palindromic (in base 2) triples that sum to n.

S000708 Number of palindromic (in base 2) quadruples that sum to n.

S000709* Lucas entry points: s(n) = smallest m >= 0 such that the n-th prime divides Lucas(m), or -1 if there is no m.

S000710 First n-digit prime in the digits of sqrt(2).

S000711 Position of the first n-digit prime in the digits of sqrt(2).

S000712 First n-digit prime in the digits of sqrt(3).

S000713 Position of the first n-digit prime in the digits of sqrt(3).

S000714 First n-digit prime in the digits of sqrt(Pi).

S000715 Position of the first n-digit prime in the digits of sqrt(Pi).

S000716 Triples of nonnegative (a,b,c) such that a*b-c, b*c-a, and c*a-b are all powers of 2.

S000717 Numbers whose Collatz (3x+1) iteration requires a different number of steps than any smaller number.

S000718 The number of Collatz (3x+1) iterations required for the n-th number in S000717.

S000719 Least prime p such that phi(f(p)) < f(p)/n, where phi is Euler’s totient function and f(p) is the product of the primes up to p.

S000720 Numbers n such that binomial(2n,n) is not divisible by 5, 7, 11, and 13.

S000721 Numbers n such that binomial(2n,n) is not divisible by 7, 11, 13, and 17.

S000722 Numbers n such that binomial(2n,n) is not divisible by 11, 13, 17, and 19.

S000723 Numbers n such that binomial(2n,n) is not divisible by 13, 17, 19, and 23.

S000724 Numbers n such that binomial(2n,n) is not divisible by 17, 19, 23, and 29.

S000725 Largest difference between increasing prime divisors of binomial(2n,n).

S000726 Difference between the the least prime >= n and the greatest prime <= 2*n/3.

S000727 Numbers n such that Lucas(n) is an abundant number.

S000728 Lucas(n) for the numbers n in S000727.

S000729 Numbers n such that Fibonacci(n) and Lucas(n) are abundant numbers.

S000730 GCD of the numbers binomial(2*n, 2*k) for k = 1..n-1.

S000731 GCD of the numbers binomial(2*prime(n), 2*k) for k = 1..prime(n)-1.

S000732 GCD of the numbers binomial(3*n, 3*k) for k = 1..n-1.

S000733 GCD of the numbers binomial(3*prime(n), 3*k) for k = 1..prime(n)-1.

S000734 GCD of the numbers binomial(4*n, 4*k) for k = 1..n-1.

S000735 GCD of the numbers binomial(4*prime(n), 4*k) for k = 1..prime(n)-1.

S000736 GCD of the numbers binomial(5*n, 5*k) for k = 1..n-1.

S000737 GCD of the numbers binomial(5*prime(n), 5*k) for k = 1..prime(n)-1.

S000738 GCD of the numbers binomial(6*n, 6*k) for k = 1..n-1.

S000739 GCD of the numbers binomial(6*prime(n), 6*k) for k = 1..prime(n)-1.

S000740 Primes p such that p = GCD of the numbers binomial(2*p, 2*k) for k = 1..p-1.

S000741 Primes p such that ((3/2)*(p-1)+1) and (3*p-2) are prime.

S000742 Pairs of numbers of the form 2^i * 3^j such that there are no primes between them.

S000743 Triangle of the number of palindromic (in bases 2 to n) triples that sum to n.

S000744 Triangle of the number of palindromic (in bases 2 to n) pairs that sum to n.

S000745 Triangle of the number of palindromic (in bases 2 to n) quadruples that sum to n.

S000746 Prime palindromes that have an increasing number of bases for which they are palindromic.

S000747 Primes p that are not palindromic in any base b with 1 < b < p-1.

S000748 The bases for which S000746(n) is palindromic.

S000749* Number of times that 2*n can be written as the sum of a prime and a member of P2.

S000750* Numbers n such that the polynomial x^n - 1 has a divisor of every degree up to n.

S000751 For each n, the highest power in each factor in the factorization of x^n - 1, sorted.

S000752 Primes p for which sum_{1..(p-1)/2} Legendre(i/p) reaches a new maximum.

S000753 For the primes p in S000752, the sum_{1..(p-1)/2} Legendre(i/p), which is a new maximum.

S000754 Primes p congruent to 1 or 5 for which sum_{1..(p-1)/4} Legendre(i/p) reaches a new maximum.

S000755 For the primes p in S000754, the sum_{1..(p-1)/4} Legendre(i/p), which is a new maximum.

S000756 Sequence of pairs (s1,s2) of the distance from a twin prime pair to the next lower and higher prime.

S000757 Prime numbers that are the sum of consecutive triangular numbers.

S000758 Least lower twin prime p such the the sum of the distances to adjacent primes is a new record.

S000759 For the lower twin primes in S000758, the total distance to the nearest primes.

S000760 For the lower twin primes in S000758, the distance to the nearest smaller prime.

S000761 For the lower twin primes in S000758, the distance to the nearest larger prime + 2.

S000762 Primes p for which the polynomial (x^11 - 1)/(x - 1) mod p is irreducible.

S000763 Primes p for which the polynomial (x^13 - 1)/(x - 1) mod p is irreducible.

S000764 Primes p for which the polynomial (x^17 - 1)/(x - 1) mod p is irreducible.

S000765 Primes p for which the polynomial (x^19 - 1)/(x - 1) mod p is irreducible.

S000766 Primes p for which the polynomial (x^23 - 1)/(x - 1) mod p is irreducible.

S000767 Primes p for which the polynomial (x^29 - 1)/(x - 1) mod p is irreducible.

S000768 Primes p for which the polynomial (x^31 - 1)/(x - 1) mod p is irreducible.

S000769 Primes p for which the polynomial (x^37 - 1)/(x - 1) mod p is irreducible.

S000770 Primes p for which the polynomial (x^41 - 1)/(x - 1) mod p is irreducible.

S000771 Primes p for which the polynomial (x^43 - 1)/(x - 1) mod p is irreducible.

S000772 Primes p for which the polynomial (x^47 - 1)/(x - 1) mod p is irreducible.

S000773 Primes p for which the polynomial (x^53 - 1)/(x - 1) mod p is irreducible.

S000774 Primes p for which the polynomial (x^59 - 1)/(x - 1) mod p is irreducible.

S000775 Primes p for which the polynomial (x^61 - 1)/(x - 1) mod p is irreducible.

S000776 Primes p for which the polynomial (x^67 - 1)/(x - 1) mod p is irreducible.

S000777 Primes p for which the polynomial (x^71 - 1)/(x - 1) mod p is irreducible.

S000778 Primes p for which the polynomial (x^73 - 1)/(x - 1) mod p is irreducible.

S000779 Primes p for which the polynomial (x^79 - 1)/(x - 1) mod p is irreducible.

S000780 Primes p for which the polynomial (x^83 - 1)/(x - 1) mod p is irreducible.

S000781 Primes p for which the polynomial (x^89 - 1)/(x - 1) mod p is irreducible.

S000782 Primes p for which the polynomial (x^97 - 1)/(x - 1) mod p is irreducible.

S000783 The first number for which there are n consecutive identical lengths of the Collatz (3X+1) iteration.

S000784 The union of the terms in S000783.

S000785 Irregular table of numbers m such that the sum (k+1)^(2n) + (k+2)^(2n) +…+ (k+m)^(2n) may be prime.

S000786 Prime numbers that are the sum of consecutive 6-th powers.

S000787 Prime numbers that are the sum of consecutive 8-th powers.

S000788 Prime numbers that are the sum of consecutive 10-th powers.

S000789 Prime numbers that are the sum of consecutive 12-th powers.

S000790 Prime numbers that are the sum of consecutive triangular numbers.

S000791 Prime numbers that are the sum of consecutive pentagonal numbers.

S000792 Prime numbers that are the sum of consecutive hexagonal numbers.

S000793 Prime numbers that are the sum of consecutive heptagonal numbers.

S000794 Prime numbers that are the sum of consecutive octagonal numbers.

S000795 Prime numbers that are the sum of consecutive 9-gonal numbers.

S000796 Prime numbers that are the sum of consecutive 10-gonal numbers.

S000797 Prime numbers that are the sum of consecutive 11-gonal numbers.

S000798 Prime numbers that are the sum of consecutive 12-gonal numbers.

S000799 Prime numbers that are the sum of consecutive 13-gonal numbers.

S000800 Record lengths of arithmetic progressions in squarefree numbers beginning with 1.

S000801 Common difference in the arithmetic progression S000800(n).

S000802 Last term in the arithmetic progression listed in S000800(n).

S000803 Triangle whose n-th row has the numbers m such that there is an m-gonal number equal to A063778(n).

S000804 Triangle in which the k-th term in n-th row has index of the S000803(n,k)-gonal number that equals A063778(n).

S000805 Floor(n^(1/4) * sqrt(log(n))).

S000806 Numbers that are not the sum of 3 generalized 7-gonal numbers.

S000807 Numbers that are not the sum of 3 generalized 8-gonal numbers.

S000808 Numbers that are not the sum of 3 generalized 9-gonal numbers.

S000809 Numbers that are not the sum of 3 generalized 10-gonal numbers.

S000810 Numbers that are not the sum of 3 generalized 11-gonal numbers.

S000811 Numbers that are not the sum of 3 generalized 12-gonal numbers.

S000812 Numbers that are not the sum of 3 generalized 13-gonal numbers.

S000813 Numbers that are not the sum of 3 generalized 14-gonal numbers.

S000814 Numbers that are not the sum of 3 generalized 15-gonal numbers.

S000815 Numbers that are not the sum of 3 generalized 16-gonal numbers.

S000816 Numbers that are not the sum of 3 generalized 17-gonal numbers.

S000817 Numbers that are not the sum of 3 generalized 18-gonal numbers.

S000818 Numbers that are not the sum of 3 generalized 19-gonal numbers.

S000819 Numbers that are not the sum of 3 generalized 20-gonal numbers.

S000820 Nondecreasing sequence of numbers whose squares (after the first term) sum to primes.

S000821 Primes formed by sequence S000820.

S000822 All linear second-order sequences are a linear combination of these two sequences.

S000823 All linear third-order sequences are a linear combination of these three sequences.

S000824 All linear fourth-order sequences are a linear combination of these four sequences.

S000825 All linear fifth-order sequences are a linear combination of these five sequences.

S000826 All linear sixth-order sequences are a linear combination of these six sequences.

S000827 All linear seventh-order sequences are a linear combination of these seven sequences.

S000828 All linear eighth-order sequences are a linear combination of these eight sequences.

S000829 All linear ninth-order sequences are a linear combination of these nine sequences.

S000830 All linear tenth-order sequences are a linear combination of these ten sequences.

S000831 All linear 11th-order sequences are a linear combination of these 11 sequences.

S000832* Sequence of sorted prime quadruples (a,b,c,d) such that a^3 + b^3 + c^3 + d^3 = 2016.

S000833 Sorted edges (pairs of vertices) on a tetrahedron.

S000834 Sorted edges (pairs of vertices a and b with a < b) on a tetrahedron.

S000835 Sorted edges (pairs of vertices) on a cube.

S000836 Sorted edges (pairs of vertices a and b with a < b) on a cube.

S000837 Sorted edges (pairs of vertices) on a octahedron.

S000838 Sorted edges (pairs of vertices a and b with a < b) on an octahedron.

S000839 Sorted edges (pairs of vertices) on a dodecahedron.

S000840 Sorted edges (pairs of vertices a and b with a < b) on a dodecahedron.

S000841 Sorted edges (pairs of vertices) on a icosahedron.

S000842 Sorted edges (pairs of vertices a and b with a < b) on an icosahedron.

S000843 Half of all regular star-polytopes having n sides.

S000844 Regular star-polytopes having n sides.

S000845 Numbers n such that the absolute value of the central term of cyclotomic(n,x) sets a new record.

S000846 Central term in the S000845(n)-th cyclotomic polynomial.

S000847* Coefficients of the 40755-th cyclotomic polynomial.

S000848 Odd sturdy numbers.

S000849 Even sturdy numbers.

S000850* Least Hamming weight of a multiple of prime(n).

S000851 Least number k such that k*prime(n) has the least Hamming weight.

S000852 Triangle in which the n-th row is the (n+1)-th prime mod the first n primes.

S000853 The number of primes among the n terms prime(n+1) mod (prime(1)..prime(n)).

S000854 Prime numbers that are not the sum of distinct squares.

S000855 The number n is the first of two consecutive primitive abundant numbers.

S000856 Each row has the n pairs of nonnegative integers whose squares sum to A000446(n).

S000857 Maximum number of primes in a number containing a total of n digits chosen from 1, 3, 7, and 9.

S000858 The least number having n digits (chosen from 1, 3, 7, and 9) containing S000857(n) primes.

S000859 Number of numbers having n digits chosen from 1, 3, 7, and 9 containing S000857(n) primes.

S000860 Least prime dividing NL(n) = sum(k=1..n) 10^(k!-1).

S000861 Least prime dividing RNL(n) = sum(k=1..n) 10^(n!-k!).

S000862 Minimal positive 5-tuple (b1,b2,b3,b4,b5) such that n = b1/b5 + b2/b1 + b3/b2 + b4/b3 + b5/b4.

S000863 Numbers n such that 5 consecutive terms of binomial(n,k) satisfy a cubic polynomial.

S000864 Irregular table containing the distinct prime factors of binomial(2*n,n).

S000865 PrimePi of the irregular table containing the distinct prime factors of binomial(2*n,n).

S000866 Conjectured number of solutions to p^i - q^j = k^2, where p=prime(n), q=prime(n+1), and k an integer.

S000867 a(n) = number of m such that sum of proper divisors of m (A001065(m)) is 2*n+1.

S000868 a(n) = number of m such that sum of proper divisors of m (A001065(m)) is 2*n.

S000869 Mersenne exponents that are part of a twin prime pair.

S000870 Mersenne exponents that are not part of a twin prime pair.

S000871 Where records occur in S000868.

S000872 Where records occur in S000867.

S000873 Difference between binomial(2*n,n) and the closest binomial(m,k) with m > 2*n and 1 < k < n-1.

S000874 Binomial(n,k) with n >= 4 and 2 <= k <= floor(n/2), sorted.

S000875 Powers n such that the sum of the digits in 2^n sets a record.

S000876 Powers n such that the sum of the digits in 3^n sets a record.

S000877 Powers n such that the sum of the digits in 4^n sets a record.

S000878 Powers n such that the sum of the digits in 5^n sets a record.

S000879 Powers n such that the sum of the digits in 6^n sets a record.

S000880 Powers n such that the sum of the digits in 7^n sets a record.

S000881 Powers n such that the sum of the digits in 8^n sets a record.

S000882 Powers n such that the sum of the digits in 9^n sets a record.

S000883* Number of times n = x^2 + y^2 + z^2 + w^2 with (x,y,z,w) integers and |x| <= |y| <= |z| <= |w|.

S000884* Number of times n = x^2 + y^2 + z^2 + w^2 with x + y + z + w a square and with (x,y,z,w) integers and |x| <= |y| <= |z| <= |w|.

S000885 Records in the number of partitions of n into 4 squares.

S000886 Where records occur in the number of partitions of n into 4 squares.

S000887 Records in the number of partitions of n into 4 squares of integers.

S000888 Where records occur in the number of partitions of n into 4 squares of integers.

S000889 Records in the number of times n = x^2 + y^2 + z^2 + w^2 with x + y + z + w a square and with (x,y,z,w) integers and |x| <= |y| <= |z| <= |w|.

S000890 Where records occur in the number of times n = x^2 + y^2 + z^2 + w^2 with x + y + z + w a square and with (x,y,z,w) integers and |x| <= |y| <= |z| <= |w|.

S000891 Pairs (m,k) with increasing m such that m is the first number having k divisors <= sqrt(m).

S000892 The numbers m from sequence S000891.

S000893 The numbers k from sequence S000891.

S000894 Pairs (m,k) with increasing m such that m is the first number having k divisors < sqrt(m).

S000895 The numbers m from sequence S000894.

S000896 The numbers k from sequence S000894.

S000897* Numbers whose reciprocals have a palindromic repeating part in base 10.

S000898* Numbers not a multiple of 10 whose reciprocals have a palindromic repeating part.

S000899 Numbers whose reciprocals have a palindromic repeating part in base 6.

S000900 Numbers not a multiple of 6 whose reciprocals have a palindromic repeating part in base 6.

S000901 Numbers whose reciprocals have a palindromic repeating part in base 7.

S000902 Numbers whose reciprocals have a palindromic repeating part in base 8.

S000903 Numbers whose reciprocals have a palindromic repeating part in base 9.

S000904 Least primordial-product number m such that there are n triples (a,b,c) with all a+b+c = k1 and all a*b*c = k2, where k1 and k2 are two constants and k1 is prime.

S000905 The primes occurring in S000904 as the constant k1.

S000906 Primordial-product numbers such that there are no triples (a,b,c) with all a+b+c = k1 and all a*b*c = k2, where k1 and k2 are two positive integers and k1 is prime.

S000907 Numbers n such that 2^n is in S000906.

S000908 For the n-th primordial-product number p, the number of triples (a,b,c) with a+b+c = k1 and a*b*c = p, where k1 is a constant that is the least possible prime (or 0 if there are no primes).

S000909 The primes (or 0) occurring in S000908 as the constant k1.

S000910 Left- or right-truncatable primes in base 9.

S000911 Left- or right-truncatable primes in base 8.

S000912 Left- or right-truncatable primes in base 7.

S000913 Left- or right-truncatable primes in base 6.

S000914 Left- or right-truncatable primes in base 5.

S000915 Left- or right-truncatable primes in base 4.

S000916 Left- or right-truncatable primes in base 3.

S000917 Number of left- or right-truncatable primes in base n.

S000918 Numbers n such that for the Gaussian integer z = n + n*I, sigma(z)/ z is a Gaussian integer.

S000919 Squarefree numbers having at least three prime factors that are in arithmetic progression.

S000920 Squarefree numbers having at least two prime factors that are in arithmetic progression.

S000921 Starting with the complex prime 1+I, the nearest complex prime x + y*I greater than the previous prime.

S000922 The difference between the real and imaginary parts of S000921.

S000923 The squared distance between adjacent points of S000921.

S000924 Squares that are the sum of two repdigit numbers.

S000925 Least number not representable as the sum of n repdigit numbers.

S000926 Least number of repdigit numbers that sum to n.

S000927 Number of ways that number n can be represented as the sum of two nonzero repdigit numbers.

S000928 Number of ways that number n can be represented as the sum of three nonzero repdigit numbers.

S000929 Number of ways that number n can be represented as the sum of four nonzero repdigit numbers.

S000930 Number of ways that number n can be represented as the sum of five nonzero repdigit numbers.

S000931 Number of ways that number n can be represented as the sum of six nonzero repdigit numbers.

S000932 Number of distinct prime factors in the n-th primitive abundant number (A006038).

S000933 Primes of the form x^2 - xy + 2y^2, but not x^2 + xy + 2y^2, with x and y nonnegative.

S000934 In base 3, these positive numbers and their squares are palindromic.

S000935 In base 4, these positive numbers and their squares are palindromic.

S000936 In base 5, these positive numbers and their squares are palindromic.

S000937 In base 6, these positive numbers and their squares are palindromic.

S000938 In base 7, these positive numbers and their squares are palindromic.

S000939 In base 8, these positive numbers and their squares are palindromic.

S000940 In base 9, these positive numbers and their squares are palindromic.

S000941 In base 10, these positive numbers and their squares are palindromic.

S000942 Record pairs (n,k) of numbers n that have a record 3x+1 (Collatz) length k, the number of terms > n.

S000945* Least twin prime p1 such that there is a smaller twin prime p2 with p1 - p2 = 2*n.

S000946 Least twin prime p2 such that there is a larger twin prime p1 with p1 - p2 = 2*n.

S000947* First-quadrant complex numbers x + y*I, sorted by norm, that are multiply-perfect.

S000948* Numbers n such that the complex number n + n*I is a complex multiply-perfect number.

S000949 Norms squared of the complex numbers in S000947.

S000950 Prime factors occurring in the norms given in S000949.

S000951 The odd integers unioned with the odd integers times 2^(2k-1) for k = 1, 2, 3,....

S000952* Sorted Diophantine quadruples; s(i)*s(j)+1 is a square for each pair s(i) and s(j), i not equal to j.

S000953 The first component of the quadruples in S000952.

S000954 The second component of the quadruples in S000952.

S000955 The third component of the quadruples in S000952.

S000956 The fourth component of the quadruples in S000952.

S000957 Decades in which 4 primes occur.

S000958 Least positive number x that begins a string of n composite numbers of the form x^2 + x + 41.

S000959 Least positive number x that begins a string of exactly n composite numbers of the form x^2 + x + 41.

S000960* Ways in which a wave can hit an n x n array of points.

S000961 Even numbers which are the sum of two primes of the form x^2 + y^2 + 1.

S000962 Even numbers which are not the sum of two primes of the form x^2 + y^2 + 1.

S000963 Numbers n such that n is in the 3x+1 (Colatz) trajectory of n+1.

S000964 Numbers n such that n-1 is in the 3x+1 (Colatz) trajectory of n.

S000965 Numbers n such that n is in the 3x+1 (Colatz) trajectory of n-1.

S000966 For the Collatz (3x+1) sequence s beginning with n, the difference between n and the next larger number in s, or 0 if there is no larger number.

S000967 For the Collatz (3x+1) sequence s beginning with n, the difference between n and the next smaller number in s.

S000968* Gaussian integers z = x + i*y, with x > 0 and |y| <= x, whose sum of divisors is z * r, where r is also a Gaussian integer.

S000969 The real and imaginary parts of sigma(z)/z for the complex numbers z in S000968.

S000970 Number of solutions to the Pell equation x^2 - d*y^2 = -1 or 1, where x is tribonacci number t(n).

S000971 Number of solutions to the Pell equation x^2 - d*y^2 = 1, where x is tribonacci number t(n).

S000972 Number of solutions to the Pell equation x^2 - d*y^2 = -1, where x is tribonacci number t(n).

S000973 Irregular table of solutions d to the Pell equation x^2 - d*y^2 = 1, where x is tribonacci number t(n).

S000974 Irregular table of solutions d to the Pell equation x^2 - d*y^2 = -1, where x is tribonacci number t(n).

S000975 Least nonnegative number k such that Lucas(k) begins with the same digits as n.

S000976 Least Lucas number that begins with the same digits as n.

S000977 Odd numbers in the terms of the central hexanomial coefficients.

S000978 Position of odd numbers in the terms of the central hexanomial coefficients.

S000979* Differences between the position of odd numbers in the terms of the central hexanomial coefficients.

S000980 Primes p such that p^2 divides k^k + (-1)^k (k-1)^(k-1) for some k > 1.

S000981 Least k > 1 such that S000980(n) divides k^k + (-1)^k (k-1)^(k-1).

S000982 Values of k in the range of 1 < k < S000980(n)^2 such that S000980(n)^2 divides k^k + (-1)^k (k-1)^(k-1).

S000983 Differences between terms in the n-th row of S000982.

S000984 Number of terms in each period of S000983.

S000985 Least k > 1 such that prime(n) divides k^k + (-1)^k (k-1)^(k-1).

S000986 Values of k with 1 < k < prime(n)^2 such that prime(n) divides k^k + (-1)^k (k-1)^(k-1).

S000987 Differences between the terms in the n-th row of S000986.

S000988 Number of terms in each period of S000987.

S000989 n^n + (-1)^n (n-1)^(n-1).

S000990 n^n - (-1)^n (n-1)^(n-1).

S000991* Pairs of numbers x > y > 0 such that x^2 + y^2 is a square, x is as small as possible, and x/y is unique.

S000992 Pairs of numbers x > y > 0 such that x^3 + y^3 is a square, x is as small as possible, and x/y is unique.

S000993 Pairs of numbers x > y > 0 such that x^5 + y^5 is a square, x is as small as possible, and x/y is unique.

S000994 Pairs of numbers x > y > 0 such that x^7 + y^7 is a square, x is as small as possible, and x/y is unique.

S000995 Pairs of numbers x > y > 0 such that x^9 + y^9 is a square, x is as small as possible, and x/y is unique.

S000996* Triples x >= y >= z that form the sides of a primitive triangle having integer area.

S000997 Triples x <= y <= z that form the sides of a primitive triangle having integer area.

S000998 Areas of the triangles given in S000996.

S000999 Integer areas of the primitive triangles having integer sides.

S001000 Triples (x,y,z) with 1 < x < y < z such that x! y! z! is a square.

S001001 Triples (x,y,z) with 1 < x < y < z and z = y + 3 such that x! y! z! is a square.

S001002 Triples (x,y,z) with 1 < x < y < z and z = y + 2 such that x! y! z! is a square.

S001003 Triples (x,y,z) with 1 < x < y < z and z = y + 1 such that x! y! z! is a square.

S001004* Least triple of palindromes x <= y <= z such that n = x + y + z.

S001005 The first number in the triple S001004(n).

S001006 Numbers n such that S001005(n) > 0.

S001007 Numbers x such that there is a primitive triangle whose sides are x, x-1, and 3.

S001008 Numbers x such that there is a primitive triangle whose sides are x, x-2, and 4.

S001009 Numbers x such that there is a primitive triangle whose sides are x, x-1, and 5.

S001010 Numbers x such that there is a primitive* triangle whose sides are x, x-2, and 6.

S001011 Numbers x such that there is a primitive triangle whose sides are x, x-1, and 7.

S001012 Numbers x such that there is a primitive triangle whose sides are x, x-2, and 8.

S001013 Numbers x such that there is a primitive triangle whose sides are x, x-1, and 9.

S001014 Numbers x such that there is a primitive* triangle whose sides are x, x-2, and 10.

S001015 Palindromes that are either non-decreasing or non-increasing toward their center.

S001016 Record Collatz (3x+1) pairs (see comments).

S001017 Numbers c from the sequence S001016.

S001018 Number of terms greater than c in the 3x+1 iteration starting with c = S001017(n).

S001019 For odd n, the number of Collatz (3x+1) iteration terms greater than n.

S001020 For even n, the number of Collatz (3x+1) iteration terms greater than n.

S001021 Smaller twin prime p such that phi(p-1) > phi(p+1), where phi is Euler’s totient function.

S001022 Smaller twin prime p such that phi(p-1) < phi(p+1), where phi is Euler’s totient function.

S001023 Smaller twin prime p such that phi(p-1) = phi(p+1), where phi is Euler’s totient function.

S001024 Smaller twin prime p such that phi(p-1) >= phi(p+1), where phi is Euler’s totient function.

S001025 Smaller twin prime p such that phi(p-1) <= phi(p+1), where phi is Euler’s totient function.

S001026 Triples 0 < a <= b < c with a + b = c, gcd(a,b) = 1, and log(c)/log(rad(a*b*c)) a new maximum.

S001027 Numbers that are the sum of cubes of four primes.

S001028 Least number that is the sum of four cubes of primes in n ways.

S001029 The n 4-tuples of primes whose cubes sum to S001028(n).

S001030 Numbers not represented by the sum of 7 positive cubes.

S001031 Prime numbers whose reverse in base 10 is also prime.

S001032 Prime numbers containing only digits 0 and 1 whose reverse in base 10 is also prime.

S001033* Numbers that are not the sum of 17 nonnegative 5-th powers.

S001034 Numbers that are not the sum of 16 nonnegative 5-th powers.

S001035 Numbers that are not the sum of 15 nonnegative 5-th powers.

S001036 Numbers that are not the sum of 14 nonnegative 5-th powers.

S001037 Numbers that are not the sum of 13 nonnegative 5-th powers.

S001038 Numbers that are not the sum of 12 nonnegative 5-th powers.

S001039 Numbers that are not the sum of 11 nonnegative 5-th powers.

S001040 Numbers that are not the sum of 10 nonnegative 5-th powers.

S001041 Numbers that are not the sum of 9 nonnegative 5-th powers.

S001042 Number of numbers between (n-1)^5 and n^5 that are not the sum of 8 nonnegative fifth powers.

S001043 Numbers that are not the sum of 8 nonnegative fifth powers.

S001044 The last 1000 numbers that are not the sum of 8 nonnegative fifth powers.

S001045 Numbers that are not the sum of 18 nonnegative 5-th powers.

S001046 Numbers that are not the sum of 19 nonnegative 5-th powers.

S001047 Numbers that are not the sum of 20 nonnegative 5-th powers.

S001048 Numbers that are not the sum of 21 nonnegative 5-th powers.

S001049 Numbers that are not the sum of 22 nonnegative 5-th powers.

S001050 Numbers that are not the sum of 23 nonnegative 5-th powers.

S001051 Numbers that are not the sum of 24 nonnegative 5-th powers.

S001052 Numbers that are not the sum of 25 nonnegative 5-th powers.

S001053 Numbers that are not the sum of 26 nonnegative 5-th powers.

S001054 Numbers that are not the sum of 27 nonnegative 5-th powers.

S001055 Number of numbers between (n-1)^5 and n^5 that are not the sum of 7 nonnegative fifth powers.

S001056 Numbers that are not the sum of 7 nonnegative fifth powers.

S001057 Last number requiring 37-n positive fifth powers to represent.

S001058 Irregular triangle of numbers requiring 37-n positive fifth powers to represent.

S001059 Number of numbers between (n-1)^5 and n^5 that are not the sum of 6 nonnegative fifth powers.

S001060 Numbers that are not the sum of 6 nonnegative fifth powers.

S001061 Primitive Pythagorean quadruples (d, c, b, a) sorted so that a <= b <= c < d.

S001062 Primitive Pythagorean quadruples (a, b, c, d) sorted so that a <= b <= c < d.

S001063 Primitive Pythagorean quintuples (a, b, c, d, e) sorted so that a <= b <= c <= d < e.

S001064 Primitive Pythagorean sextuples (a, b, c, d, e, f) sorted.

S001065 Primitive Pythagorean septuples (a, b, c, d, e, f, g) sorted.

S001066 Primitive Pythagorean octuples (a, b, c, d, e, f, g, h) sorted.

S001067 Primitive Pythagorean 9-tuples (a, b, c, d, e, f, g, h, i) sorted.

S001068 Primitive Pythagorean 10-tuples (a, b, c, d, e, f, g, h, i, j) sorted.

S001069 Primitive Pythagorean 11-tuples (a, b, c, d, e, f, g, h, i, j, k) sorted.

S001070 Numbers n such that the digits of prime(1) to prime(n) concatenated in some order is possibly a palindrome.

S001071 Numbers n such that the digits of 1 to n concatenated in some order is possibly a palindrome.

S001072 The last 1000 numbers that are not the sum of fourth powers of distinct primes.

S001073 Number of numbers between prime(n)^4 and prime(n+1)^4 that are not the sum of distinct fourth powers of primes.

S001074 Smallest positive number s(n) such that s(n) and all greater numbers can be expressed as the sum of n-th powers of distinct primes.

S001075 For each odd prime p, the p least-multiples of p whose prime factors are in arithmetic progression.

S001076 For each odd prime p, the p common differences of the arithmetic progressions mentioned in S001075.

S001077 Numbers n for which a record number of numbers binomial(n,m) are squarefree, where m <= n/2.

S001078 The number of numbers for which binomial(n,m) is squarefree, where n is in S001077 and m <= n/2.

S001079 Irregular triangle of numbers m for which binomial(S001077(n),m) is squarefree.

S001080 Triangular table in which the n-th row has the n numbers that are the larger of a pair of numbers x and y such that x^2 - y^2 = A094191(n).

S001081 Triangular table in which the n-th row has the n numbers that are the smaller of a pair of numbers x and y such that x^2 - y^2 = A094191(n).

S001082 Numbers n such that binomial(2n,n) is fourth-power free.

S001083 Numbers n such that binomial(2n,n) is fifth-power free.

If any of these sequences are used in some manner, the page describing the sequence must be referenced. For example:

T. D. Noe, Sequence S000687, Integer Sequences, (www.IntegerSequences.org/s000687.html).

The sequences were not part of the OEIS database (www.oeis.org) when they were added to this website. Contact: noe (at) sspectra.com

* nice sequences