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

0,4

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

Formula

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

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 *)

Crossrefs

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

Sequence in context: A274638 A146244 A146457 * A353765 A358584 A003536

Adjacent sequences: A051041 A051042 A051043 * A051045 A051046 A051047

Keyword

nonn

Author

Eric W. Weisstein

Extensions

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

Status

approved