A232504 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).

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, 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, 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

Offset: 1

Author

Zhi-Wei Sun, Nov 25 2013

Keywords

• nonn

Comments

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

Examples

			a(5) = 1 since 5 = 1 + 4 with p(1) + q(4) = 1 + 2 = 3 prime.
a(8) = 1 since 8 = 4 + 4 with p(4) + q(4) = 5 + 2 = 7 prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Z.-W. Sun, On a^n+ bn modulo m, arXiv preprint arXiv:1312.1166 [math.NT], 2013-2014.

Crossrefs

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

Programs

Mathematica

a[n_]:=Sum[If[PrimeQ[PartitionsP[k]+PartitionsQ[n-k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]