A228431 Number of ordered ways to write n = x + y (x, y > 0) with p(3, x) + p(6, y) prime, where p(3, k) denotes the triangular number k*(k+1)/2 and p(6, k) denotes the hexagonal number k*(2*k-1) = p(3, 2*k-1).

0, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 4, 4, 1, 4, 3, 1, 2, 3, 2, 5, 5, 3, 2, 3, 2, 4, 3, 3, 8, 4, 1, 3, 2, 2, 11, 5, 1, 5, 5, 4, 4, 5, 4, 7, 4, 3, 7, 6, 3, 9, 4, 2, 5, 4, 3, 12, 7, 2, 4, 10, 1, 7, 8, 4, 10, 7, 3, 10, 9, 5, 8, 5, 4, 10, 9, 5, 10, 9, 3, 12, 13, 4, 4, 9, 4, 11, 10, 5, 11, 16, 5, 10, 8, 5, 16, 8, 3, 11, 15

Offset: 1

Author

Zhi-Wei Sun, Nov 10 2013

Keywords

• nonn

Comments

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

This implies that there are infinitely many primes each of which can be written as a sum of a triangular number and a hexagonal number.

See also A228425, A228428, A228429 and A228430 for more similar conjectures.

Examples

			a(14) = 1 since 14 = 10 + 4 with p(3, 10) + p(6, 4) = 83 prime.
a(38) = 1 since 38 = 31 + 7 with p(3, 31) + p (6, 7) = 587 prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A000217, A000384, A228425, A228428, A228429, A228430.

Programs

Mathematica

p[m_,x_]:=(m-2)x(x-1)/2+x
a[n_]:=Sum[If[PrimeQ[p[3,x]+p[6,n-x]],1,0],{x,1,n-1}]
Table[a[n],{n,1,100}]