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%I #6 Feb 27 2014 03:52:56
%S 0,0,1,2,2,3,2,3,3,2,3,2,5,3,5,4,4,3,4,4,6,2,4,3,5,2,4,1,4,5,6,5,5,4,
%T 5,3,4,3,5,6,5,6,3,8,6,5,6,4,6,7,5,6,4,6,7,6,7,7,6,6,7,5,6,5,6,5,5,5,
%U 7,7,6,5,7,9,8,6,5,5,7,6,8,6,5,8,7,8,7,4,8,7,7,7,6,6,6,6,6,7,6,9
%N Number of primes p < n with q(n-p) + 1 prime, where q(.) is the strict partition function (A000009).
%C Conjecture: a(n) > 0 for all n > 2. Also, for each n > 6 there is a prime p < n with q(n-p) - 1 prime.
%C We have verified the conjecture for n up to 10^5.
%C See also A238458 for a similar conjecture involving the partition function p(n).
%H Zhi-Wei Sun, <a href="/A238459/b238459.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(3) = 1 since 2 and q(3-2) + 1 = 1 + 1 = 2 are both prime.
%e a(28) = 1 since 17 and q(28-17) + 1 = q(11) + 1 = 12 + 1 = 13 are both prime.
%t q[n_,k_]:=PrimeQ[PartitionsQ[n-Prime[k]]+1]
%t a[n_]:=Sum[If[q[n,k],1,0],{k,1,PrimePi[n-1]}]
%t Table[a[n],{n,1,100}]
%Y Cf. A000009, A000040, A237705, A237768, A237769, A238457, A238458.
%K nonn
%O 1,4
%A _Zhi-Wei Sun_, Feb 27 2014
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