cp's OEIS Frontend

A238516 a(n) = |{0 < k < n: (p(k)+1)*p(n) + 1 is prime}|, where p(.) is the partition function (A000041).

%I A238516 #12 Feb 28 2014 02:57:01
%S A238516 0,1,1,1,2,3,3,3,4,5,5,2,4,6,5,3,3,3,4,1,7,7,2,6,3,8,7,4,1,6,3,4,5,8,
%T A238516 4,4,2,2,4,9,7,6,3,6,4,2,6,6,3,8,5,6,4,7,7,4,8,7,9,1,6,7,7,3,3,7,2,5,
%U A238516 4,10,8,5,1,8,9,1,4,6,7,12,3,2,4,10,4,4
%N A238516 a(n) = |{0 < k < n: (p(k)+1)*p(n) + 1 is prime}|, where p(.) is the partition function (A000041).
%C A238516 Conjecture: (i) a(n) > 0 for all n > 1. Also, for any integer n > 4 there is a positive integer k < n with (p(k)-1)*p(n) - 1 prime.
%C A238516 (ii) Let q(.) be the strict partition function (A000009). If n > 5, then p(n)*q(k) + 1 is prime for some 3 < k < n. If n > 6, then p(n)*q(k) - 1 is prime for some 0 < k < n. If n > 1, then q(n)*q(k) + 1 is prime for some 0 < k < n. If n > 3, then q(n)*q(k) - 1 is prime for some 0 < k < n.
%C A238516 We have verified that a(n) > 0 for all n = 2, 3, ..., 60000.
%H A238516 Zhi-Wei Sun, <a href="/A238516/b238516.txt">Table of n, a(n) for n = 1..10000</a>
%H A238516 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014.
%e A238516 a(4) = 1 since (p(1)+1)*p(4) + 1 = 2*5 + 1 = 11 is prime.
%e A238516 a(20) = 1 since (p(12)+1)*p(20) + 1 = 78*627 + 1 = 48907 is prime.
%e A238516 a(246) = 1 since (p(45)+1)*p(246) + 1 = 89135*169296722391554 + 1 = 15090263350371165791 is prime.
%t A238516 p[n_,k_]:=PrimeQ[PartitionsP[n]*(PartitionsP[k]+1)+1]
%t A238516 a[n_]:=Sum[If[p[n,k],1,0],{k,1,n-1}]
%t A238516 Table[a[n],{n,1,100}]
%Y A238516 Cf. A000009, A000040, A000041, A233417, A238457, A238509.
%K A238516 nonn
%O A238516 1,5
%A A238516 _Zhi-Wei Sun_, Feb 28 2014