S000763


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

2, 7, 11, 13, 19, 37, 41, 59, 67, 71, 89, 97, 137, 149, 163, 167, 193, 197, 223, 227, 241, 271, 293, 331, 349, 353, 379, 383, 397, 401, 409, 431, 449, 457, 461, 479, 487, 509, 557, 587, 613, 617, 631, 643, 661, 683, 691, 709, 739, 743, 761, 769, 773, 787

1

S000763

This is the 12th-degree cyclotomic polynomial. These primes p satisfy the congruence p mod 13 == {0, 2, 6, 7, 11}.  The fraction of all primes in this sequence is 1/3. If we plotted n versus primepi(s(n)), then the plotted points would be very close to the line having slope 3.

T. D. Noe, Plot of 1000 terms

T. D. Noe, Table of 1000 terms

Wikipedia, Cyclotomic polynomial

(Mma) t = {}; n = 13; p = 1; While[Length[t] < 100, p = NextPrime[p]; If[Length[FactorList[(x^n - 1)/(x - 1), Modulus -> p]] == 2, AppendTo[t, p]]]; t

Cf. A045309A042993A045401, S000762-S000782.

nonn

T. D. Noe, Dec 01 2015

© Tony D Noe 2014-2015