S000640

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

0, 12673, 41788, 47656, 82348, 85485, 95325, 109020, 144313, 165385, 171465, 207417, 215373, 227268, 251781, 258013, 271660, 303025, 316848, 357717, 396340, 407092, 410533, 436356, 464145, 490360, 503217, 529720, 533281, 544453, 585156, 600117, 603748, 644368

1

Note that 3478727 = 7 * 17 * 23 * 31 * 41, the product of the first 5 primes of the form 8k+-1 which are in A001132. Sequences for the product of the first 3 and 4 of these primes are in A201916 and A201917. The terms satisfy an order 487 linear difference equation (given below).

T. D. Noe, Plot of 1000 terms

T. D. Noe, Table of 1000 terms

Formua: s(n) = s(n-1) + 6*s(n-243) - 6*s(n-244) - s(n-486) + s(n-487), where the 487 initial terms can be computed using the Mathematica program.

(Mma) d = 3478727  terms = 1000; t = Select[Range[0, 69574540 , IntegerQ[Sqrt[#^2 + (#+d)^2]] &]; Do[AppendTo[t, t[[-1]] + 6*t[[-243]] - 6*t[[-244]] - t[[-486]] + t[[-487]]], {terms-487}]; t

Cf. A001132A201916, A201917.

nonn

T. D. Noe, May 19 2015