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

1,1

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

Table of n, a(n) for n=1..50.

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A158075 A087594 A320868 * A319167 A088909 A340919

Adjacent sequences: A347813 A347814 A347815 * A347817 A347818 A347819

Keyword

nonn

Author

Sela Fried, Sep 15 2021

Status

approved