A114913 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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.

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 *)

CROSSREFS

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

A111174 is a subsequence.

See comments in A113780 for explanation.

Sequence in context: A083317 A268514 A024514 * A111174 A075751 A065153

Adjacent sequences: A114910 A114911 A114912 * A114914 A114915 A114916

KEYWORD

nonn

AUTHOR

Christian G. Bower, Jan 06 2006

STATUS

approved