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A347816 Prime numbers p such that both 15 and 85 are quadratic nonresidue (mod p).

%I A347816 #26 Sep 16 2021 11:46:34
%S A347816 13,29,31,41,47,79,83,139,157,199,211,263,269,373,379,383,401,433,439,
%T A347816 443,449,457,467,499,521,563,571,577,587,613,619,641,647,691,733,751,
%U A347816 757,809,811,821,863,881,929,937,941,991,1033,1049,1051,1061
%N A347816 Prime numbers p such that both 15 and 85 are quadratic nonresidue (mod p).
%C A347816 Primes p such that E_6(x)/(x + 1) is irreducible (mod p) where E_6(x) is the Eulerian polynomial and E_6(x)/(x + 1) = x^4 + 56x^3 + 246x^2 + 56x + 1. (See A159041.)
%C A347816 The sequence is infinite.
%C A347816 It is the intersection of A038888 and A038972.
%H A347816 A. J. J. Heidrich, <a href="https://doi.org/10.1016/0022-314X(84)90050-7">On the factorization of Eulerian polynomials</a>, Journal of Number Theory, 18(2):157-168, 1984.
%p A347816 alias(ls = NumberTheory:-LegendreSymbol):
%p A347816 isA347816 := k -> isprime(k) and ls(15, k) = -1 and ls(85, k) = -1:
%p A347816 A347816List := upto -> select(isA347816, [`$`(3..upto)]):
%p A347816 A347816List(1061); # _Peter Luschny_, Sep 16 2021
%t A347816 Select[Prime@Range[180], JacobiSymbol[15, #] == -1 && JacobiSymbol[85,#]==-1 &] (* _Stefano Spezia_, Sep 16 2021 *)
%o A347816 (PARI) isok(p) = isprime(p) && (kronecker(15,p)==-1) && (kronecker(85,p)==-1); \\ _Michel Marcus_, Sep 16 2021
%o A347816 (Python)
%o A347816 from sympy.ntheory import legendre_symbol, primerange
%o A347816 A347816_list = [p for p in primerange(3,10**5) if legendre_symbol(15,p) == legendre_symbol(85,p) == -1] # _Chai Wah Wu_, Sep 16 2021
%Y A347816 Cf. A159041, A008292, A173018, A038888, A038972, A347815.
%K A347816 nonn
%O A347816 1,1
%A A347816 _Sela Fried_, Sep 15 2021