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

1,1

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

Links

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

Example

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

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<p such that # p+q and p-q are both oblong numbers. isA162652 := 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 *)

Crossrefs

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

Sequence in context: A308851 A298027 A085104 * A306889 A181141 A031158

Adjacent sequences: A162649 A162650 A162651 * A162653 A162654 A162655

Keyword

nonn

Author

Daniel Tisdale, Jul 08 2009

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

Status

approved