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

1,1

Comments

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

Links

Table of n, a(n) for n=1..52.

Example

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

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 *)

Crossrefs

Cf. A082612.

Sequence in context: A064098 A181078 A265773 * A059784 A000283 A121910

Adjacent sequences: A098714 A098715 A098716 * A098718 A098719 A098720

Keyword

nonn

Author

Robin Garcia, Sep 29 2004

Extensions

More terms from Zak Seidov, Feb 16 2005

Status

approved