%I #16 Mar 25 2020 06:04:59
%S 7,13,31,43,73,211,241,421,463,1123,1723,2551,2971,4831,5701,6163,
%T 8011,8191,9901,11131,12433,14281,17293,19183,20023,23563,24181,28393,
%U 30103,31153,35911,37831,43891,46441,53593,60271,77563,83233,86143,95791
%N 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.
%C 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
%C Also, primes of the form (p^2+3)/4 with p odd prime. - _Zak Seidov_, May 10 2014
%H Jean-François Alcover, <a href="/A162652/b162652.txt">Table of n, a(n) for n = 1..77</a>
%e 7 = 1^2+1+5 = 3^2+3-5.
%p 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
%t sol[p_] := m^2 + m - p /. Solve[m>0 && n>0 && 2p == m + m^2 + n + n^2, {m, n}, Integers];
%t 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 *)
%Y Cf. A163418. - _R. J. Mathar_, Feb 05 2010
%K nonn
%O 1,1
%A _Daniel Tisdale_, Jul 08 2009
%E Definition revised by _N. J. A. Sloane_, Jul 17 2009
%E More terms from _R. J. Mathar_, Jul 17 2009
%E Extended beyond a(31) by _R. J. Mathar_, Feb 05 2010