A129541 Primes of the form p^2 + q^2 + A007918(p + q) - p - q, where p and q are consecutive primes.

13, 37, 2333, 51229, 84131, 141539, 273821, 591893, 649813, 744221, 889877, 911269, 1065829, 2146619, 2205013, 2766007, 2913773, 3090187, 3348893, 3374821, 3505979, 3942493, 4095547, 4885981, 5766421, 6125029, 6336829

Offset: 1

Author

J. M. Bergot, Jun 08 2007

Keywords

• nonn

Comments

The trial number was 80 pairs of consecutive primes to produce seven primes. Oddly it seems more productive as the pairs of primes increase in value, rather unusual for generators of primes. Perhaps an extension will confirm this.

Examples

			Take consecutive primes 31 and 37. The sum 31 + 37 = 68 and is three less than the next prime 71. Hence 31^2 + 37^2 + 3 = 961+1369+3=2333 which is a prime that belongs to the sequence.
		

Programs

Mathematica

cp[{a_,b_}]:=a^2+b^2+NextPrime[a+b]-a-b; Join[{13},Select[cp/@ Partition[ Prime[Range[500]],2,1],PrimeQ]] (* Harvey P. Dale, Nov 16 2013 *)

PARI

p=2;forprime(q=3,1e4,t=p^2+q^2+nextprime(p+q)-p-q;if(isprime(t),print1(t","));p=q)

Formula

For two consecutive primes p and q, add them and subtract that amount from the nearest prime greater than p+q. Call this number d; then see whether p^2 + q^2 + d is a prime.

Extensions

Extended, edited, and program added by Charles R Greathouse IV, Nov 11 2009