A227899 Number of primes p < n with 3*p - 4 and n^2 + (n - p)^2 both prime.

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

Offset: 1

Author

Zhi-Wei Sun, Oct 14 2013

Keywords

• nonn

Comments

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

Examples

			a(5) = 1 since 5 = 3 + 2, and the three numbers 3, 3*3 - 4 = 5 and 5^2 + (5-3)^2 = 29 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. A069003, A185636, A204065, A227898, A230223.

Programs

Mathematica

a[n_]:=Sum[If[PrimeQ[3Prime[i]-4]&&PrimeQ[n^2+(n-Prime[i])^2],1,0],{i,1,PrimePi[n-1]}]
Table[a[n],{n,1,100}]