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A051044 Odd values of the PartitionsQ function A000009.

%I A051044 #38 Feb 16 2025 08:32:41
%S A051044 1,1,1,3,5,15,27,89,165,585,1113,4097,7917,29927,58499,225585,444793,
%T A051044 1741521,3457027,13699699,27342421,109420549,219358315,884987529,
%U A051044 1780751883,7233519619,14600965705,59656252987,120742510607,495811828759,1005862035461
%N A051044 Odd values of the PartitionsQ function A000009.
%C A051044 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
%C A051044 _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)."
%C A051044 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
%H A051044 Alois P. Heinz, <a href="/A051044/b051044.txt">Table of n, a(n) for n = 0..600</a>
%H A051044 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PartitionFunctionQCongruences.html">Partition Function Q Congruences</a>
%F A051044 a(n) = A000009(A001318(n)). - _Reinhard Zumkeller_, Apr 22 2006
%p A051044 b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
%p A051044      `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
%p A051044     end:
%p A051044 a:= n-> b((m->m*(3*m-1)/2)(ceil(-n*(-1)^n/2))):
%p A051044 seq(a(n), n=0..30);  # _Alois P. Heinz_, Aug 23 2021
%t A051044 PartitionsQ /@ Table[n*((n + 1)/6), {n, Select[Range[50], Mod[#, 3] != 1 & ]}] (* _Jean-François Alcover_, Oct 31 2012, after _Reinhard Zumkeller_ *)
%Y A051044 Cf. A000009, A001318, A035359, A051005, A118303.
%K A051044 nonn
%O A051044 0,4
%A A051044 _Eric W. Weisstein_
%E A051044 Missing initial 1 inserted by _Sean A. Irvine_, Aug 23 2021