A236193 Primes p with prime(p)^2 + (2*p)^2 and p^2 + (2*prime(p))^2 both prime.

3, 139, 179, 233, 491, 929, 1217, 1429, 1597, 1613, 1987, 2243, 3061, 3499, 3529, 4507, 5737, 5779, 6329, 7247, 7823, 8263, 8839, 9941, 10259, 11317, 11383, 12157, 12421, 13093, 13219, 13367, 14449, 14669, 15101, 15877, 17449, 18523, 18593, 19051

Offset: 1

Author

Zhi-Wei Sun, Jan 20 2014

Keywords

• nonn

Comments

By part (i) of the conjecture in A236192, this sequence should have infinitely many terms.

Examples

			a(1) = 3 since prime(2)^2 + (2*2)^2 = 25 is composite, but prime(3)^2 + (2*3)^2 = 5^2 + 6^2 = 61 and 3^2 + (2*prime(3))^2 = 3^2 + 10^2 = 109 are both prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A236119, A236143, A236192.

Programs

Mathematica

p[n_]:=PrimeQ[Prime[n]^2+(2*n)^2]&&PrimeQ[n^2+(2*Prime[n])^2]
n=0;Do[If[p[Prime[k]],n=n+1;Print[n," ",Prime[k]]],{k,1,10^6}]