A114913 Numbers k such that A114912(k) = 1. Numbers k such that A000009(k) == 2 (mod 4).

3, 4, 8, 10, 13, 14, 17, 18, 19, 24, 25, 28, 32, 39, 42, 43, 47, 48, 50, 52, 54, 55, 62, 67, 69, 73, 74, 75, 76, 78, 83, 84, 87, 88, 89, 90, 95, 99, 101, 103, 105, 108, 109, 112, 113, 118, 119, 123, 125, 127, 130, 132, 134, 138, 140, 143, 144, 147, 149, 153, 154, 157

Offset: 1

Author

Christian G. Bower, Jan 06 2006

Keywords

• nonn

Comments

All the terms are the sum of a generalized pentagonal number A001318 and a square A000290.

Let 24*k+1 = p_1^e_1 * ... * p_r^e_r * q_1^f_1 * ... * q_s^f_s, where the p_i's are distinct primes == 1, 5, 7, or 11 (mod 24) and the q_i's are distinct primes == 13, 17, 19, or 23 (mod 24). Then k belongs to the sequence iff all of the f_i's are even and all but one of the e_i's are even and the one e_i which is odd is == 1 (mod 4). - Dean Hickerson, Jan 19 2006

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Krishnaswami Alladi, Partition Identities Involving Gaps and Weights, Transactions of the American Mathematical Society, Vol. 349, No. 12 (Dec 1997), pp. 5001-5019.

Crossrefs

Cf. A000009, A000290, A001318, A114912, A114914.

A111174 is a subsequence.

See comments in A113780 for explanation.

Programs

Mathematica

q[n_] := Module[{f = FactorInteger[n], f1, f2}, f1 = Select[f, MemberQ[{1, 5, 7, 11}, Mod[First[#], 24]] &]; f2 = Select[f, MemberQ[{13, 17, 19, 23}, Mod[First[#], 24]] &]; AllTrue[f2[[;;, 2]], EvenQ] && Count[f1[[;;, 2]], ?OddQ] == 1]; Select[Range[160], q[24 * # + 1] &] (* _Amiram Eldar, Aug 24 2024 *)