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A239232 a(n) = |{0 < k <= n: p(n+k) + 1 is prime}|, where p(.) is the partition function (A000041).

%I A239232 #8 Mar 13 2014 11:15:36
%S A239232 1,0,0,1,3,3,3,2,1,1,1,1,2,4,4,4,4,4,3,4,4,4,4,4,4,3,2,2,2,3,3,3,3,3,
%T A239232 4,4,4,4,4,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,3,4,5,5,4,4,4,4,4,4,5,5,6,6,
%U A239232 7,6,6,6,6,7,8,8,9,9,9,10
%N A239232 a(n) = |{0 < k <= n: p(n+k) + 1 is prime}|, where p(.) is the partition function (A000041).
%C A239232 Conjecture: (i) a(n) > 0 for all n > 3.
%C A239232 (ii) If n > 15, then p(n+k) - 1 is prime for some k = 1, ..., n.
%C A239232 (iii) If n > 38, then p(n+k) is prime for some k = 1, ..., n.
%C A239232 The conjecture implies that there are infinitely many positive integers m with p(m) + 1 (or p(m) - 1, or p(m)) prime.
%H A239232 Zhi-Wei Sun, <a href="/A239232/b239232.txt">Table of n, a(n) for n = 1..10000</a>
%H A239232 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014.
%e A239232 a(4) = 1 since p(4+4) + 1 = 22 + 1 = 23 is prime.
%e A239232 a(8) = 2 since p(8+1) + 1 = 31 and p(8+2) + 1 = 43 are both prime.
%e A239232 a(11) = 1 since p(11+8) + 1 = 491 is prime.
%t A239232 p[n_]:=PartitionsP[n]
%t A239232 a[n_]:=Sum[If[PrimeQ[p[n+k]+1],1,0],{k,1,n}]
%t A239232 Table[a[n],{n,1,80}]
%Y A239232 Cf. A000040, A000041, A049575, A234470, A234569, A238393, A238457, A238509, A238516, A239207, A239209, A239214.
%K A239232 nonn
%O A239232 1,5
%A A239232 _Zhi-Wei Sun_, Mar 13 2014