A228428 Number of ways to write n = x + y (x, y > 0) with p(99, x) + p(100, 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, 1, 3, 3, 1, 2, 4, 1, 3, 5, 4, 3, 5, 1, 4, 3, 2, 1, 7, 5, 1, 4, 4, 4, 4, 5, 2, 5, 4, 2, 6, 1, 4, 9, 6, 4, 6, 6, 5, 6, 5, 4, 6, 5, 11, 9, 6, 5, 10, 4, 3, 9, 5, 3, 11, 7, 7, 10, 5, 5, 10, 5, 5, 5, 4, 7, 6, 6, 5, 13, 7, 3, 12, 5, 5, 12, 6, 7, 8, 6, 4, 9, 7, 8, 9, 10, 9, 9, 8, 15, 17, 9, 9, 10

Offset: 1

Author

Zhi-Wei Sun, Nov 10 2013

Keywords

• nonn

Comments

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

See also A228425 for other similar conjectures.

Examples

			a(8) = 1 since 8 = 3 + 5 with p(99, 3) + p (100, 5) = 1279 prime.
a(38) = 1 since 38 = 6 + 32 with p(99, 6) + p(100, 32) = 50101 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.

Programs

Mathematica

p[m_,x_]:=(m-2)x(x-1)/2+x
a[n_]:=Sum[If[PrimeQ[p[99,x]+p[100,n-x]],1,0],{x,1,n-1}]
Table[a[n],{n,1,100}]