cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

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

Original entry on oeis.org

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

Views

Author

Jianing Song, Sep 05 2018

Keywords

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.

Examples

			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).

Links

Crossrefs

Cf. A318527.

Programs

Formula

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

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

Original entry on oeis.org

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

Views

Author

Jianing Song, Aug 06 2018

Keywords

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

Showing 1-2 of 2 results.

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