A130761 Primes prime(n) such that at least one of the two numbers (prime(n+2)^2-prime(n)^2)/2 - 1 and (prime(n+2)^2-prime(n)^2)/2 + 1 is prime.

3, 5, 7, 11, 13, 19, 29, 31, 37, 41, 43, 53, 59, 61, 67, 71, 79, 83, 97, 107, 127, 139, 149, 157, 179, 181, 191, 197, 227, 229, 239, 251, 263, 283, 293, 307, 347, 349, 353, 373, 419, 439, 443, 463, 467, 479, 499, 523, 541, 569, 601, 607, 613, 617, 619

Offset: 1

Author

J. M. Bergot, Jul 13 2007

Keywords

• nonn
• less

Examples

			(7^2 - 3^2)/2 - 1 is 19. Therefore 3 is in the sequence.
(19^2 - 13^2)/2 + 1 is 97. Hence 13 is in the sequence.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Programs

Maple

Res:= NULL:
p:= 5: q:= 3:
count:= 0:
while count < 100 do
  r:= q; q:= p; p:= nextprime(p);
  v:= (p^2-r^2)/2;
  if isprime(v+1) or isprime(v-1) then
    count:= count+1; Res:= Res, r;
  fi
od:
Res; # Robert Israel, Oct 03 2018

Mathematica

Prime[Select[Range[140], PrimeQ[(Prime[ #+2]^2-Prime[ # ]^2)/2+1] || PrimeQ[(Prime[ # +2]^2-Prime[ # ]^2)/2-1] &]]
Select[Partition[Prime[Range[200]],3,1],AnyTrue[(#[[3]]^2-#[[1]]^2)/2+{1,-1},PrimeQ]&][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 28 2020 *)

Extensions

Edited and extended by Stefan Steinerberger, Jul 23 2007