%I
%S 0,0,1,2,2,3,2,3,4,2,3,3,3,5,2,4,4,5,4,5,4,4,3,3,3,4,4,4,2,4,2,5,4,4,
%T 5,3,3,6,3,4,1,3,4,7,6,4,4,4,4,4,4,5,3,5,5,7,3,3,4,6,5,8,5,5,4,4,2,7,
%U 5,7,3,6,5,7,6,7,5,5,4,7,4,5,3,5,6,8,5,3,4,6,3,5,4,5,4,5,2,6,4,5
%N Number of primes p < n with 2*P(np) + 1 prime, where P(.) is the partition function (A000041).
%C Conjecture: a(n) > 0 for all n > 2. Also, for each n > 3 there is a prime p < n with 2*P(np)  1 prime.
%C We have verified the conjecture for n up to 10^5.
%C See also A238459 for a similar conjecture involving the strict partition function.
%H ZhiWei Sun, <a href="/A238458/b238458.txt">Table of n, a(n) for n = 1..10000</a>
%H Z.W. 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 2*P(32) + 1 = 2*1 + 1 = 3 are both prime.
%e a(41) = 1 since 37 and 2*P(4137) + 1 = 2*5 + 1 = 11 are both prime.
%t p[n_,k_]:=PrimeQ[2*PartitionsP[nPrime[k]]+1]
%t a[n_]:=Sum[If[p[n,k],1,0],{k,1,PrimePi[n1]}]
%t Table[a[n],{n,1,100}]
%Y Cf. A000040, A000041, A237705, A237768, A237769, A238457, A238459.
%K nonn
%O 1,4
%A _ZhiWei Sun_, Feb 27 2014
