A228429 Number of ways to write n = x + y (x, y > 0) with p(39, x) + p(40, y) prime, where p(m, k) denotes the m-gonal number (m-2)*k*(k-1)/2 + k.

0, 1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 3, 1, 1, 4, 2, 2, 4, 1, 3, 1, 2, 5, 5, 1, 2, 3, 4, 3, 11, 4, 4, 2, 3, 4, 9, 6, 3, 5, 6, 3, 5, 4, 3, 9, 6, 3, 4, 7, 5, 13, 8, 3, 5, 5, 6, 13, 9, 9, 6, 3, 4, 6, 7, 3, 5, 5, 8, 5, 11, 8, 11, 8, 5, 10, 9, 5, 13, 9, 10, 11, 4, 7, 14, 12, 6, 11, 9, 5, 9, 15, 5, 14, 11, 6, 7, 8, 13, 14

Offset: 1

Author

Zhi-Wei Sun, Nov 10 2013

Keywords

• nonn

Comments

By a conjecture in A228425, we should have a(n) > 0 for all n > 1.

Conjecture: For each m = 3, 4, ..., any sufficiently large integer n can be written as x + y (x, y > 0) with p(m, x) + p(m+1, y) prime.

Examples

			a(9) = 1 since 9 = 5 + 4 with p(39, 5) + p(40, 4) = 607 prime.
a(26) = 1 since 26 = 19 + 7 with p(39, 19) + p (40, 7) = 7151 prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, On universal sums of polygonal numbers, preprint, arXiv:0905.0635 [math.NT], 2009-2015.

Crossrefs

Cf. A000040, A228425, A228428, A228430, A228431.

Programs

Mathematica

p[m_,x_]:=(m-2)x(x-1)/2+x
a[n_]:=Sum[If[PrimeQ[p[39,x]+p[40,n-x]],1,0],{x,1,n-1}]
Table[a[n],{n,1,100}]