A098717 Primes p such that 2*p+1 and ((2*p+1)^2 + 1)/2 = p^2 + (p+1)^2 are primes.

2, 5, 29, 719, 1229, 1409, 1559, 2039, 2399, 2699, 2939, 3449, 3779, 6269, 6899, 7079, 8069, 9689, 12959, 13619, 14009, 14249, 14879, 19559, 20369, 20759, 21089, 22079, 22469, 23459, 26879, 28559, 30269, 31799, 32009, 32789, 33179, 33569, 38639, 39989, 40949, 41399, 41969, 42359, 45569, 46349, 47279, 49499, 49919, 53309, 54959, 55469

Offset: 1

Author

Robin Garcia, Sep 29 2004

Keywords

• nonn

Comments

It is easy to prove that all the terms except the first two must satisfy a(n) mod 10 = 9.

Examples

			a(3) = 29 = p and 2*p + 1 = 59 and (59^2 + 1)/2 = 29^2 + 30^2 = 1741 are prime.
		

Crossrefs

Cf. A082612.

Programs

Mathematica

Flatten[Append[{2, 5}, Select[Sort[Range[29, 30000000, 30], Range[49, 30000000, 30]], PrimeQ[ # ]&&PrimeQ[2 # + 1] && PrimeQ[1 + 2 # + 2 #^2] &]]] (Zak Seidov)
f1[n_]:=(n+1)^2-n^2;f2[n_]:=(n+1)^2+n^2; Select[Prime[Range[8! ]],PrimeQ[f1[ # ]]&&PrimeQ[f2[ # ]]&] (* Vladimir Joseph Stephan Orlovsky, Feb 25 2010 *)

Extensions

More terms from Zak Seidov, Feb 16 2005