A227899 - OEIS

%I #11 Oct 15 2013 03:10:28

%S 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,

%T 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,

%U 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

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

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

%H Zhi-Wei Sun, <a href="/A227899/b227899.txt">Table of n, a(n) for n = 1..10000</a>

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, preprint, arXiv:1211.1588.

%e 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.

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

%t Table[a[n],{n,1,100}]

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

%K nonn

%O 1,8

%A _Zhi-Wei Sun_, Oct 14 2013