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%I #12 Mar 14 2014 13:47:20
%S 1,2,3,2,3,1,4,5,2,4,5,4,4,4,2,4,3,6,3,1,3,5,5,5,2,9,8,7,5,3,3,4,3,7,
%T 4,8,6,2,6,6,5,2,5,5,3,3,4,4,7,7,8,5,5,4,8,6,3,4,3,5,11,2,2,4,6,6,5,5,
%U 4,4,5,6,6,8,4,9,4,6,4,3
%N a(n) = |{0 <= k < n: q(n+k*(k+1)/2) + 1 is prime}|, where q(.) is the strict partition function given by A000009.
%C We note that a(n) > 0 for n up to 3580 with the only exception n = 1831. Also, for n = 722, there is no number k among 0, ..., n with q(n+k(k+1)/2) - 1 prime.
%H Zhi-Wei Sun, <a href="/A239238/b239238.txt">Table of n, a(n) for n = 1..600</a>
%e a(6) = 1 since q(6+0*1/2) + 1 = q(6) + 1 = 5 is prime.
%e a(20) = 1 since q(20+8*9/2) + 1 = q(56) + 1 = 7109 is prime.
%e a(104) = 1 since q(104+15*16/2 + 1 = q(224) + 1 = 1997357057 is prime.
%e a(219) = 1 since q(219+65*66/2) + 1 = q(2364) + 1 = 111369933847869807268722580000364711 is prime.
%e a(1417) > 0 since q(1417+1347*1348/2) + 1 = q(909295) + 1 is prime.
%t q[n_]:=PartitionsQ[n]
%t a[n_]:=Sum[If[PrimeQ[q[n+k(k+1)/2]+1],1,0],{k,0,n-1}]
%t Table[a[n],{n,1,80}]
%Y Cf. A000009, A000040, A000217, A185636, A239232.
%K nonn
%O 1,2
%A _Zhi-Wei Sun_, Mar 13 2014
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