A227898 Number of primes p < n with p + 6 and n + (n - p)^2 both prime.

0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 3, 1, 4, 3, 2, 2, 3, 3, 3, 3, 3, 4, 4, 2, 2, 3, 3, 3, 2, 2, 5, 5, 2, 5, 4, 2, 4, 5, 2, 7, 5, 3, 4, 5, 3, 3, 4, 4, 3, 5, 4, 9, 9, 2, 5, 3, 4, 8, 6, 2, 5, 8, 3, 4, 7, 3, 10, 5, 2, 7, 4, 5, 10, 6, 4, 6, 6, 2, 6, 8, 3, 6, 5, 3, 6, 6, 5, 9, 4, 5, 7, 5, 4, 9, 10

Offset: 1

Author

Zhi-Wei Sun, Oct 14 2013

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n > 5.

(ii) For any integer n > 5, there is a prime p with p + 6 and n*(n - p) - 1 both prime.

Examples

			a(6) = 1 since 5, 5 + 6 = 11 and 6 + (6 - 5)^2 = 7 are all 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. A023201, A185636, A204065, A227899, A230261.

Programs

Mathematica

a[n_]:=Sum[If[PrimeQ[Prime[i]+6]&&PrimeQ[n+(n-Prime[i])^2],1,0],{i,1,PrimePi[n-1]}]
Table[a[n],{n,1,100}]