A162652 Primes p such that there are positive integers m and n and a prime q such that p = m^2+m-q = n^2+n+q.

7, 13, 31, 43, 73, 211, 241, 421, 463, 1123, 1723, 2551, 2971, 4831, 5701, 6163, 8011, 8191, 9901, 11131, 12433, 14281, 17293, 19183, 20023, 23563, 24181, 28393, 30103, 31153, 35911, 37831, 43891, 46441, 53593, 60271, 77563, 83233, 86143, 95791

Offset: 1

Author

Daniel Tisdale, Jul 08 2009

Keywords

• nonn

Comments

To test if a prime p is a member, p = n^2+n+q gives a finite list of possible pairs (n,q), and, for each value of q, m^2+m = p+q determines a putative value of m. - N. J. A. Sloane, Jul 17 2009

Also, primes of the form (p^2+3)/4 with p odd prime. - Zak Seidov, May 10 2014

Examples

			7 = 1^2+1+5 = 3^2+3-5.
		

Links

• Jean-François Alcover, Table of n, a(n) for n = 1..77

Crossrefs

Cf. A163418. - R. J. Mathar, Feb 05 2010

Programs

Maple

isA002378 := proc(n) if n >= 0 then if issqr(4*n+1) then RETURN(type( sqrt(4*n+1),'odd')) ; else false; fi; else false; fi; end: # primes p such there is a prime q
A162652 := proc(p) local j,q; if isprime(p) then for j from 1 do q := ithprime(j) ; if q >= p then break; fi; if isA002378(p+q) and isA002378(p-q) then RETURN(true) ; fi; od: false ; else false; fi; end: for n from 1 to 4000 do if isA162652(ithprime(n)) then printf("%d,",ithprime(n)) ; fi; od; # R. J. Mathar, Jul 17 2009

Mathematica

sol[p_] := m^2 + m - p /. Solve[m>0 && n>0 && 2p == m + m^2 + n + n^2, {m, n}, Integers];
Reap[For[p = 2, p < 10^6, p = NextPrime[p], qsel = Select[sol[p], PrimeQ]; If[qsel != {}, Print[p]; Sow[p]]]][[2, 1]] (* Jean-François Alcover, Mar 25 2020 *)

Extensions

Definition revised by N. J. A. Sloane, Jul 17 2009

More terms from R. J. Mathar, Jul 17 2009

Extended beyond a(31) by R. J. Mathar, Feb 05 2010