A229166 Number of ordered ways to write n = x*(x+1)/2 + y with y*(y+1)/2 + 1 prime, where x and y are nonnegative integers.

1, 1, 1, 3, 1, 1, 3, 2, 2, 2, 3, 2, 3, 3, 1, 2, 3, 3, 3, 2, 2, 5, 3, 2, 2, 4, 2, 2, 4, 2, 2, 2, 3, 1, 3, 2, 3, 2, 2, 4, 3, 1, 3, 5, 2, 3, 4, 5, 2, 4, 2, 3, 3, 2, 3, 5, 4, 2, 4, 1, 4, 3, 5, 4, 3, 5, 3, 4, 3, 3, 6, 4, 2, 5, 4, 3, 4, 5, 5, 2, 4, 4, 2, 3, 6, 4, 2, 3, 5, 4, 3, 5, 1, 4, 3, 6, 3, 5, 7, 3

Offset: 1

Author

Zhi-Wei Sun, Oct 15 2013

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 0. Moreover, if n > 0 is not among 1, 3, 60, then there are positive integers x and y with x*(x+1)/2 + y = n such that y*(y+1)/2 + 1 is prime.

Examples

			a(6) = 1 since 6 = 2*3/2 + 3 with 3*4/2 + 1 = 7 prime.
a(60) = 1 since 60 = 0*1/2 + 60 with 60*61/2 + 1 = 1831 prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, On sums of primes and triangular numbers, J. Comb. Number Theory 1(2009), 65-76.
• Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588 [math.NT], 2012-2017.

Crossrefs

Cf. A000040, A000217, A132399, A208244, A230121, A230252.

Programs

Mathematica

T[n_]:=n(n+1)/2
a[n_]:=Sum[If[PrimeQ[T[n-T[i]]+1],1,0],{i,0,(Sqrt[8n+1]-1)/2}]
Table[a[n],{n,1,100}]