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

1,8

Comments

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

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.

Example

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.

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}]

Crossrefs

Cf. A069003, A185636, A204065, A227898, A230223.

Sequence in context: A284322 A219607 A205451 * A208244 A272171 A160696

Adjacent sequences: A227896 A227897 A227898 * A227900 A227901 A227902

Keyword

nonn

Author

Zhi-Wei Sun, Oct 14 2013

Status

approved