A235614 Number of ordered ways to write n = k + m with k a term of A235592 and m a positive triangular number.

0, 0, 1, 1, 2, 2, 2, 2, 3, 2, 2, 4, 1, 3, 3, 2, 3, 3, 3, 3, 5, 2, 3, 5, 3, 3, 3, 2, 4, 6, 2, 4, 3, 2, 4, 4, 4, 2, 6, 4, 4, 6, 2, 5, 2, 3, 7, 5, 4, 4, 6, 1, 2, 6, 5, 4, 5, 4, 5, 5, 1, 4, 7, 5, 5, 4, 2, 3, 5, 4, 4, 8, 4, 6, 4, 4, 4, 1, 2, 4, 7, 5, 3, 5, 3, 5, 3, 2, 6, 6, 4, 6, 8, 1, 4, 5, 5, 4, 7, 6

Offset: 1

Author

Zhi-Wei Sun, Jan 13 2014

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 2.

Examples

			a(13) = 1 since 13 = 3 + 10 with 3*4 - prime(3) = 7 prime and 10 = 4*5/2 a positive triangular number.
a(52) = 1 since 52 = 37 + 15 with 37*38 - prime(37) = 1249 prime and 15 = 5*6/2 a positive triangular number.
a(61) = 1 since 61 = 6 + 55 with 6*7 - prime(6) = 29 prime and 55 = 10*11/2 a positive triangular number.
a(313) = 1 since 313 = 37 + 276 with 37*38 - prime(37) = 1249 prime and 276 = 23*24/2 a positive triangular number.
		

Links

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

Crossrefs

Cf. A000040, A000217, A235592, A235613.

Programs

Mathematica

PQ[n_]:=PrimeQ[n(n+1)-Prime[n]]
TQ[n_]:=IntegerQ[Sqrt[8n+1]]
a[n_]:=Sum[If[PQ[k]&&TQ[n-k],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]