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Odd values of the PartitionsQ function A000009.
4

%I #37 Aug 24 2021 17:05:44

%S 1,1,1,3,5,15,27,89,165,585,1113,4097,7917,29927,58499,225585,444793,

%T 1741521,3457027,13699699,27342421,109420549,219358315,884987529,

%U 1780751883,7233519619,14600965705,59656252987,120742510607,495811828759,1005862035461

%N Odd values of the PartitionsQ function A000009.

%C 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 _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 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 Alois P. Heinz, <a href="/A051044/b051044.txt">Table of n, a(n) for n = 0..600</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PartitionFunctionQCongruences.html">Partition Function Q Congruences</a>

%F a(n) = A000009(A001318(n)). - _Reinhard Zumkeller_, Apr 22 2006

%p b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(

%p `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)

%p end:

%p a:= n-> b((m->m*(3*m-1)/2)(ceil(-n*(-1)^n/2))):

%p seq(a(n), n=0..30); # _Alois P. Heinz_, Aug 23 2021

%t PartitionsQ /@ Table[n*((n + 1)/6), {n, Select[Range[50], Mod[#, 3] != 1 & ]}] (* _Jean-François Alcover_, Oct 31 2012, after _Reinhard Zumkeller_ *)

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

%K nonn

%O 0,4

%A _Eric W. Weisstein_

%E Missing initial 1 inserted by _Sean A. Irvine_, Aug 23 2021