A051044 Odd values of the PartitionsQ function A000009.

1, 1, 1, 3, 5, 15, 27, 89, 165, 585, 1113, 4097, 7917, 29927, 58499, 225585, 444793, 1741521, 3457027, 13699699, 27342421, 109420549, 219358315, 884987529, 1780751883, 7233519619, 14600965705, 59656252987, 120742510607, 495811828759, 1005862035461

Offset: 0

Author

Eric W. Weisstein

Keywords

• nonn

Comments

A000009(n) is odd iff n is of the form k*(3*k - 1)/2 or k*(3*k + 1)/2. - Jonathan Vos Post, Jun 18 2005

Eric W. Weisstein comments: "The values of n for which A000009(n) is prime are 3, 4, 5, 7, 22, 70, 100, 495, 1247, 2072, 320397, ... (A035359). These values correspond to 2, 2, 3, 5, 89, 29927, 444793, 602644050950309, ... (A051005). It is not known if a(n) is infinitely often prime, but Gordon and Ono (1997) proved that it is 'almost always' divisible by any given power of 2 (1997)."

Semiprime values begin: a(5) = 15 = 3 * 5, a(11) = 4097 = 17 * 241, a(20) = 27342421 = 389 * 70289, a(24) = 1780751883 = 3 * 593583961, a(28) = 120742510607 = 31 * 3894919697. - Jonathan Vos Post, Jun 18 2005

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..600
• Eric Weisstein's World of Mathematics, Partition Function Q Congruences

Crossrefs

Cf. A000009, A001318, A035359, A051005, A118303.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= n-> b((m->m*(3*m-1)/2)(ceil(-n*(-1)^n/2))):
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 23 2021

Mathematica

PartitionsQ /@ Table[n*((n + 1)/6), {n, Select[Range[50], Mod[#, 3] != 1 & ]}] (* Jean-François Alcover, Oct 31 2012, after Reinhard Zumkeller *)

Formula

a(n) = A000009(A001318(n)). - Reinhard Zumkeller, Apr 22 2006

Extensions

Missing initial 1 inserted by Sean A. Irvine, Aug 23 2021