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

13, 29, 31, 41, 47, 79, 83, 139, 157, 199, 211, 263, 269, 373, 379, 383, 401, 433, 439, 443, 449, 457, 467, 499, 521, 563, 571, 577, 587, 613, 619, 641, 647, 691, 733, 751, 757, 809, 811, 821, 863, 881, 929, 937, 941, 991, 1033, 1049, 1051, 1061

Offset: 1

Author

Sela Fried, Sep 15 2021

Keywords

• nonn

Comments

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.)

The sequence is infinite.

It is the intersection of A038888 and A038972.

Links

• A. J. J. Heidrich, On the factorization of Eulerian polynomials, Journal of Number Theory, 18(2):157-168, 1984.

Crossrefs

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

Programs

Maple

alias(ls = NumberTheory:-LegendreSymbol):
isA347816 := k -> isprime(k) and ls(15, k) = -1 and ls(85, k) = -1:
A347816List := upto -> select(isA347816, [`$`(3..upto)]):
A347816List(1061); # Peter Luschny, Sep 16 2021

Mathematica

Select[Prime@Range[180], JacobiSymbol[15, #] == -1 && JacobiSymbol[85,#]==-1 &] (* Stefano Spezia, Sep 16 2021 *)

PARI

isok(p) = isprime(p) && (kronecker(15,p)==-1) && (kronecker(85,p)==-1); \\ Michel Marcus, Sep 16 2021

Python

from sympy.ntheory import legendre_symbol, primerange
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