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%I #41 Apr 06 2017 21:28:00
%S 2,3,5,7,11,13,17,23,37,43,47,53,67,73,97,103,107,113,127,137,157,163,
%T 167,173,193,223,233,257,263,277,283,293,313,337,347,353,373,397,433,
%U 443,467,487,523,547,563,577,593,607,613,617,643,647,653,733,743,757,773,787,797,887,907,937,947,953,977
%N Primes of the form (p^2 - q^2) / 24 with primes p > q > 3.
%C Note that p - q must be <= 12. Also note that there can be corresponding prime pairs (q, p) more than one way, i.e., (7, 13), (13, 17), (29, 31): (13^2 - 7^2)/24 = (17^2 - 13^2)/24 = (31^2 - 29^2)/24 = 5.
%C There are no terms of A045468 > 11.
%C Union of {2}, A006489, A060212, A092110, and A125272. - _Robert Israel_, Mar 13 2017
%H Robert Israel, <a href="/A283562/b283562.txt">Table of n, a(n) for n = 1..10000</a>
%F For n > 5, a(n) == {3,7} mod 10.
%e 3 is a term since (11^2 - 7^2)/24 = 3 and 3, 7, 11 are prime numbers.
%p select(r -> isprime(r) and ((isprime(3*r+2) and isprime(3*r-2))
%p or (isprime(6*r+1) and isprime(6*r-1))
%p or (isprime(2*r+3) and isprime(2*r-3))
%p or (isprime(r+6) and isprime(r-6))), [2,seq(i,i=3..1000,2)]); # _Robert Israel_, Mar 13 2017
%t ok[n_] := PrimeQ[n] && Block[{p, q, s = Reduce[p^2-q^2 == 24 n && p>3 && q>3, {p, q}, Integers]}, If[s === {}, False, Or @@ And @@@ PrimeQ[{p, q} /. List@ ToRules@s]]]; Select[Range@1000, ok] (* _Giovanni Resta_, Mar 11 2017 *)
%o (PARI) isA124865(n) = if(n%24, isprimepower(n+4)==2 || isprimepower(n+9)==2, fordiv(n/4, d, if(isprime(n/d/4+d) && isprime(n/d/4-d), return(1))); 0)
%o lista(nn) = forprime(p=2, nn, if(isA124865(24*p), print1(p", ")))
%Y Cf. A006489, A060212, A092110, A124865, A124866, A125272, A283532.
%K nonn
%O 1,1
%A _Altug Alkan_ and _Thomas Ordowski_, Mar 11 2017
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