A187785 Number of ways to write n=x+y (x,y>=0) with {6x-1,6x+1} a twin prime pair and y a triangular number.

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

Offset: 1

Author

Zhi-Wei Sun, Jan 06 2013

Keywords

• nonn

Comments

Conjecture: a(n)>0 for all n>48624 not equal to 76106.

We have verified this for n up to 2*10^8. It seems that 723662 is the unique n>76106 which really needs y=0 in the described representation.

Compare the conjecture with another Sun's conjecture associated with A132399.

Examples

			a(9)=1 since 9=3+3(3+1)/2 with 6*3-1 and 6*3+1 both prime.
		

References

• Zhi-Wei Sun, On sums of primes and triangular numbers, J. Comb. Number Theory 1(2009), no. 1, 65-76.

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..100000
• 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, arXiv:1211.1588 [math.NT], 2012-2017.

Crossrefs

Cf. A001097, A000217, A132399, A187759, A187757, A219157.

Programs

Mathematica

a[n_]:=a[n]=Sum[If[PrimeQ[6(n-k(k+1)/2)-1]==True&&PrimeQ[6(n-k(k+1)/2)+1]==True,1,0],{k,0,(Sqrt[8n+1]-1)/2}]
Do[Print[n," ",a[n]],{n,1,100}]