A377476 - OEIS

%I #82 Nov 18 2025 13:22:03

%S 479,1151,1319,1559,2351,2689,2999,3529,3671,3911,4751,4919,5519,5569,

%T 5711,6551,6599,7559,7561,7681,8089,8761,8951,9239,9241,9601,9719,

%U 9769,10391,10559,10799,12049,12239,12721,12911,13151,13729,14159,14281,14759,14951,15671,15791,16631,16921

%N Primes p such that 1..12 are quadratic residues modulo p.

%C An odd prime p is a term if and only if the Legendre symbol Legendre(q|p) = 1 for all q = 2,3,5,7,11; i.e., each prime q <= 12 is a quadratic residue.

%C Prime p is a term if and only if all the following conditions are satisfied:

%C   p == +-1 (mod 24)

%C   p == +-1 (mod 10)

%C   p == +-1, +-3, +-9 (mod 28)

%C   p == +-1, +-5, +-7, +-9, +-19 (mod 44)

%C Prime p is a term if and only if it is congruent to any number in the attached file modulo 9240.

%H Andrew Howroyd, <a href="/A377476/b377476.txt">Table of n, a(n) for n = 1..10000</a>

%H Steven Lu, <a href="/A377476/a377476_1.txt">Congruence classes modulo 9240 of the terms</a>

%e 479 is a term of this sequence, since Legendre(b|479) = 1 for b = 1, 2, ..., 12.

%t Select[Prime /@ Range[2000], And @@ Table[KroneckerSymbol[b, #] == 1, {b, Range[12]}] &]

%o (PARI) isok(p)={for(i=1, 12, if(kronecker(i,p)<0, return(0))); isprime(p)} \\ _Andrew Howroyd_, Feb 17 2025

%K nonn

%O 1,1

%A _Steven Lu_, Feb 16 2025