Primes p for which the polynomial (x^53 - 1)/(x - 1) mod p is irreducible.
2, 3, 5, 19, 31, 41, 53, 61, 67, 71, 73, 79, 101, 103, 109, 127, 137, 139, 151, 157, 167, 173, 179, 181, 191, 193, 233, 239, 251, 257, 263, 277, 283, 313, 337, 349, 353, 359, 373, 379, 383, 389, 397, 419, 421, 443, 457, 463, 479, 491, 499, 503, 509, 557
1
This is the 52nd-degree cyclotomic polynomial. These primes p satisfy the congruence p mod 53 == {0, 2, 3, 5, 8, 12, 14, 18, 19, 20, 21, 22, 26, 27, 31, 32, 33, 34, 35, 39, 41, 45, 48, 50, 51}. The fraction of all primes in this sequence is 6/13. If we plotted n versus primepi(s(n)), then the plotted points would be very close to the line having slope 13/6.
T. D. Noe, Plot of 1000 terms
T. D. Noe, Table of 1000 terms
Wikipedia, Cyclotomic polynomial
(Mma) t = {}; n = 53; 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. A045309, A042993, A045401, S000762-S000782.
nonn
T. D. Noe, Dec 01 2015