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

1,2

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.

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, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Example

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

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}]

Crossrefs

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

Sequence in context: A197169 A048052 A184156 * A238277 A258757 A351399

Adjacent sequences: A187782 A187783 A187784 * A187786 A187787 A187788

Keyword

nonn

Author

Zhi-Wei Sun, Jan 06 2013

Status

approved