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

1,1

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.

Links

Table of n, a(n) for n=1..27.

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.

Example

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.

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 *)

Prog

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

Crossrefs

Sequence in context: A262475 A309808 A201809 * A302858 A244185 A044090

Adjacent sequences: A129538 A129539 A129540 * A129542 A129543 A129544

Keyword

nonn

Author

J. M. Bergot, Jun 08 2007

Extensions

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

Status

approved