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A263721 The prime p in the Fouvry-Iwaniec prime k^2 + p^2 (A185086), or the larger of k and p if both are prime.

%I A263721 #18 Aug 21 2017 13:09:18
%S A263721 2,3,5,5,7,5,3,5,3,7,11,7,11,13,7,2,13,13,5,17,13,11,5,17,7,17,19,3,
%T A263721 17,7,19,5,11,19,13,23,5,17,19,13,23,5,2,19,17,11,5,23,29,29,23,19,29,
%U A263721 13,31,31,23,11,3,5,31,13,2,29,5,13,31,2,11,5,31,37,23,37,3,7,23,3,13,31,19,37,41,11
%N A263721 The prime p in the Fouvry-Iwaniec prime k^2 + p^2 (A185086), or the larger of k and p if both are prime.
%C A263721 The sequence is well-defined by the uniqueness part of Fermat's two-squares theorem.
%C A263721 The sequence is infinite, since Fouvry and Iwaniec proved that A185086 is infinite.
%H A263721 Charles R Greathouse IV, <a href="/A263721/b263721.txt">Table of n, a(n) for n = 1..10000</a>
%H A263721 Étienne Fouvry and Henryk Iwaniec, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa79/aa7935.pdf">Gaussian primes</a>, Acta Arithmetica 79:3 (1997), pp. 249-287.
%F A263721 a(n)^2 = A185086(n) - k^2 for some integer k > 0.
%e A263721 A185086(2) = 13 = 2^2 + 3^2 and A185086(6) = 61 = 5^2 + 6^2, so a(2) = 3 and a(6) = 5.
%t A263721 p = 2; lst = {}; While[p < 100, k = 1; While[k < 101, If[PrimeQ[k^2 + p^2], AppendTo[lst, {k^2 + p^2, If[PrimeQ@ k, Max[k, p], p]}]]; k++]; p = NextPrime@ p]; Transpose[Union@ lst][[2]]
%o A263721 (PARI) do(lim)=my(v=List(),p2,t); forprime(p=2, sqrtint(lim\=1), p2=p^2; for(k=1, sqrtint(lim-p2), if(isprime(t=p2+k^2), listput(v, [t, if(isprime(k),max(k,p),p)])))); v=vecsort(Set(v),1); apply(u->u[2], v) \\ _Charles R Greathouse IV_, Aug 21 2017
%Y A263721 Cf. A002313, A002144, A028916, A045637, A062324, A185086, A240130, A262340, A263722.
%K A263721 nonn
%O A263721 1,1
%A A263721 _Jonathan Sondow_ and _Robert G. Wilson v_, Oct 24 2015