A305864 Numbers k such that -3 is a quadratic residue (not necessarily coprime) modulo k, k + 1 and k + 2.

1, 2, 12, 37, 146, 156, 181, 217, 397, 541, 721, 722, 732, 876, 937, 1082, 1092, 1117, 1226, 1236, 1261, 1442, 1621, 1657, 1812, 1981, 2017, 2197, 2306, 2316, 2557, 2676, 2917, 3061, 3097, 3252, 3396, 3457, 3601, 3612, 3746, 3781, 3962, 3997, 4106, 4177, 4357, 4501

Offset: 1

Author

Jianing Song, Aug 06 2018

Keywords

• nonn

Comments

Start of 3 consecutive terms in A057128. All terms are congruent to {1, 2} mod 5 and {0, 1, 2} mod 6.

If k is a term of this sequence then -3 is a quadratic residue modulo k*(k + 1)*(k + 2)/2, but the converse is not true if k, (k + 1)/2 and k + 2 are terms in A057128 and k == 7 (mod 16) (k = 487, 631, 2071, ...).

Examples

			12 is a term since 3^2 == -3 (mod 12), 6^2 == -3 (mod 13) and 5^2 == -3 (mod 14).
		

Crossrefs

Cf. A057128.

Cf. A318527 (start of 4 consecutive terms in A057128).

Programs

PARI

isA057128(n) = issquare(Mod(-3, n));
isA305864(n) = isA057128(n)&&isA057128(n+1)&&isA057128(n+2);
for(n=1, 10000, if(isA305864(n), print1(n, ", ")))