A230254 Number of ways to write n = p + q with p and (p+1)*q/2 + 1 both prime.

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

Offset: 1

Author

Zhi-Wei Sun, Oct 14 2013

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 3.

We have verified this for n up to 10^8.

We also have some similar conjectures, for example, any integer n > 3 not equal to 17 or 66 can be written as p + q with p and (p+1)*q/2 - 1 both prime.

Examples

			a(15) = 1 since 15 = 5 + 10 with 5 and (5+1)*10/2+1 = 31 both prime.
a(30) = 1 since 30 = 2 + 28 with 2 and (2+1)*28/2+1 = 43 both prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.

Crossrefs

Cf. A000040, A002375, A219864, A230252, A227908, A227909, A230241.

Programs

Mathematica

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