A347816 - OEIS

%I #26 Sep 16 2021 11:46:34

%S 13,29,31,41,47,79,83,139,157,199,211,263,269,373,379,383,401,433,439,

%T 443,449,457,467,499,521,563,571,577,587,613,619,641,647,691,733,751,

%U 757,809,811,821,863,881,929,937,941,991,1033,1049,1051,1061

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

%C 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 The sequence is infinite.

%C It is the intersection of A038888 and A038972.

%H 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 alias(ls = NumberTheory:-LegendreSymbol):

%p isA347816 := k -> isprime(k) and ls(15, k) = -1 and ls(85, k) = -1:

%p A347816List := upto -> select(isA347816, [`$`(3..upto)]):

%p A347816List(1061); # _Peter Luschny_, Sep 16 2021

%t Select[Prime@Range[180], JacobiSymbol[15, #] == -1 && JacobiSymbol[85,#]==-1 &] (* _Stefano Spezia_, Sep 16 2021 *)

%o (PARI) isok(p) = isprime(p) && (kronecker(15,p)==-1) && (kronecker(85,p)==-1); \\ _Michel Marcus_, Sep 16 2021

%o (Python)

%o from sympy.ntheory import legendre_symbol, primerange

%o 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 Cf. A159041, A008292, A173018, A038888, A038972, A347815.

%K nonn

%O 1,1

%A _Sela Fried_, Sep 15 2021