A318911 Numbers k such that -3 is a quadratic residue modulo 360*k + 1, 360*k + 2, 360*k + 3 and 360*k + 4.

0, 2, 13, 17, 18, 20, 21, 25, 31, 40, 47, 51, 54, 57, 68, 69, 76, 83, 91, 102, 109, 110, 117, 119, 120, 131, 132, 134, 138, 142, 145, 149, 168, 171, 174, 176, 179, 182, 183, 189, 204, 205, 207, 208, 211, 212, 218, 229, 230, 234, 245, 253, 263, 281, 286, 293, 295

Offset

1,2

Comments

Companion sequence to A318527, as it is shown there that all terms in A318527 are congruent to 1 mod 360.

Also numbers k such that -3 is a quadratic residue modulo (360*k + 1)*(360*k + 2)*(360*k + 3)*(360*k + 4)/2.

The number of terms <= 1000, 10000 and 100000 are 156, 1100 and 8056, respectively. There are also 22 pairs of consecutive numbers <= 1000, 99 pairs <= 10000 and 540 pairs <= 100000.

Links

Altug Alkan, Table of n, a(n) for n = 1..10000

Formula

a(n) = (A318527(n) - 1)/360.

Example

2 is a term since 93^2 == -3 (mod 721), 137^2 == -3 (mod 722), 210^2 == -3 (mod 723) and 97^2 == -3 (mod 724).

Prog

(PARI) isA057128(n) = issquare(Mod(-3, n));
isA318911(n) = isA057128(360*n+1) && isA057128(360*n+2) && isA057128(360*n+3) && isA057128(360*n+4);

Crossrefs

Cf. A318527.

Sequence in context: A298295 A257636 A258317 * A065245 A158845 A163786

Adjacent sequences: A318908 A318909 A318910 * A318912 A318913 A318914

Keyword

nonn

Author

Jianing Song, Sep 05 2018

Status

approved