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A347816
Prime numbers p such that both 15 and 85 are quadratic nonresidue (mod p).
1
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
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
KEYWORD
nonn
AUTHOR
Sela Fried, Sep 15 2021
STATUS
approved