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%I #10 Feb 27 2014 22:35:12
%S 0,1,1,2,2,3,2,1,1,2,1,2,3,4,1,4,5,2,1,5,2,1,1,3,5,2,3,2,2,4,7,3,2,2,
%T 5,6,3,7,3,3,4,3,3,2,2,4,7,4,8,3,9,4,6,4,3,7,3,2,3,4,5,3,7,4,3,5,1,9,
%U 10,6,8,2,3,3,6,6,3,1,2,7,1,6,5,2,6,8,3,4,1,1,1,9,12,3,2,3,8,4,3,2
%N a(n) = |{0 < k < n: p(n) + p(k) - 1 is prime}|, where p(.) is the partition function (A000041).
%C Conjecture: (i) a(n) > 0 for all n > 1. Also, if n > 2 is different from 8 and 25, then p(n) + p(k) + 1 is prime for some 0 < k < n.
%C (ii) If n > 7, then n + p(k) is prime for some 0 < k < n.
%C (iii) If n > 1, then p(k) + q(n) is prime for some 0 < k < n, where q(.) is the strict partition function given by A000009. If n > 2, then p(k) + q(n) - 1 is prime for some 0 < k < n. If n > 1 is not equal to 8, then p(k) + q(n) + 1 is prime for some 0 < k < n.
%H Zhi-Wei Sun, <a href="/A238509/b238509.txt">Table of n, a(n) for n = 1..10000</a>
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014.
%e a(11) = 1 since p(11) + p(10) - 1 = 56 + 42 - 1 = 97 is prime.
%e a(247) = 1 since p(247) + p(228) - 1 = 182973889854026 + 40718063627362 - 1 = 223691953481387 is prime.
%t p[n_,k_]:=PrimeQ[PartitionsP[n]+PartitionsP[k]-1]
%t a[n_]:=Sum[If[p[n,k],1,0],{k,1,n-1}]
%t Table[a[n],{n,1,100}]
%Y Cf. A000009, A000040, A000041, A232504, A238457.
%K nonn
%O 1,4
%A _Zhi-Wei Sun_, Feb 27 2014
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