A305864 - OEIS

%I #59 May 26 2019 19:38:44

%S 1,2,12,37,146,156,181,217,397,541,721,722,732,876,937,1082,1092,1117,

%T 1226,1236,1261,1442,1621,1657,1812,1981,2017,2197,2306,2316,2557,

%U 2676,2917,3061,3097,3252,3396,3457,3601,3612,3746,3781,3962,3997,4106,4177,4357,4501

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

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

%C 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, ...).

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

%o (PARI) isA057128(n) = issquare(Mod(-3, n));

%o isA305864(n) = isA057128(n)&&isA057128(n+1)&&isA057128(n+2);

%o for(n=1, 10000, if(isA305864(n), print1(n, ", ")))

%Y Cf. A057128.

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

%K nonn

%O 1,2

%A _Jianing Song_, Aug 06 2018