This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A129541 #9 Nov 16 2013 11:03:14
%S A129541 13,37,2333,51229,84131,141539,273821,591893,649813,744221,889877,
%T A129541 911269,1065829,2146619,2205013,2766007,2913773,3090187,3348893,
%U A129541 3374821,3505979,3942493,4095547,4885981,5766421,6125029,6336829
%N A129541 Primes of the form p^2 + q^2 + A007918(p + q) - p - q, where p and q are consecutive primes.
%C A129541 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.
%F A129541 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.
%e A129541 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.
%t A129541 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 *)
%o A129541 (PARI) p=2;forprime(q=3,1e4,t=p^2+q^2+nextprime(p+q)-p-q;if(isprime(t),print1(t","));p=q)
%K A129541 nonn
%O A129541 1,1
%A A129541 _J. M. Bergot_, Jun 08 2007
%E A129541 Extended, edited, and program added by _Charles R Greathouse IV_, Nov 11 2009