A232504 - OEIS

%I #12 Aug 05 2019 09:26:13

%S 0,1,2,2,1,1,4,1,5,4,5,4,4,3,5,5,6,2,4,8,4,3,6,5,3,5,5,8,5,6,4,7,5,5,

%T 2,6,9,8,3,10,7,9,7,4,7,8,8,5,6,8,5,4,8,5,5,7,11,7,7,9,8,7,9,11,8,10,

%U 4,7,8,7,9,13,7,8,4,6,11,8,13,3,8,10,5,7,11,11,6,9,6,5,10,6,9,5,10,11,9,8,11,8

%N Number of ways to write n = k + m (k, m > 0) with p(k) + q(m) prime, where p(.) is the partition function (A000041) and q(.) is the strict partition function (A000009).

%C Conjecture: a(n) > 0 for all n > 1.

%H Zhi-Wei Sun, <a href="/A232504/b232504.txt">Table of n, a(n) for n = 1..10000</a>

%H Z.-W. Sun, <a href="http://arxiv.org/abs/1312.1166">On a^n+ bn modulo m</a>, arXiv preprint arXiv:1312.1166 [math.NT], 2013-2014.

%e a(5) = 1 since 5 = 1 + 4 with p(1) + q(4) = 1 + 2 = 3 prime.

%e a(8) = 1 since 8 = 4 + 4 with p(4) + q(4) = 5 + 2 = 7 prime.

%t a[n_]:=Sum[If[PrimeQ[PartitionsP[k]+PartitionsQ[n-k]],1,0],{k,1,n-1}]

%t Table[a[n],{n,1,100}]

%Y Cf. A000009, A000040, A000041, A202650, A231201.

%K nonn

%O 1,3

%A _Zhi-Wei Sun_, Nov 25 2013