A238393 - OEIS

%I #18 Mar 01 2014 09:52:02

%S 1,1,3,2,3,2,5,2,6,3,3,5,1,9,3,4,5,5,6,2,7,3,5,8,3,4,8,10,7,10,6,7,9,

%T 8,8,6,6,4,12,10,10,8,6,6,5,7,8,6,10,5,9,9,11,7,7,6,9,11,8,7,11,6,9,8,

%U 4,8,5,18,14,10,9,7,8,6,13,9,4,7,7,15

%N a(n) = |{0 < k <= n: 2*p(k)*p(n) + 1 is prime}|, where p(.) is the partition function (A000041).

%C Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 2, 13.

%C (ii) If n > 1, then 2*p(k)*p(n) - 1 is prime for some 0 < k < n.

%C (iii) For any integer n > 0, p(k)*(p(n)+1) + 1 is prime for some k = 1, ..., n.

%H Zhi-Wei Sun, <a href="/A238393/b238393.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(2) = 1 since 2*p(1)*p(2) + 1 = 2*1*2 + 1 = 5 is prime.

%e a(13) = 1 since 2*p(3)*p(13) + 1 = 2*3*101 + 1 = 607 is prime.

%t p[n_,k_]:=PrimeQ[2*PartitionsP[n]*PartitionsP[k]+1]

%t a[n_]:=Sum[If[p[n,k],1,0],{k,1,n}]

%t Table[a[n],{n,1,80}]

%Y Cf. A000040, A000041, A238457, A238509, A238516.

%K nonn

%O 1,3

%A _Zhi-Wei Sun_, Mar 01 2014