%I
%S 5,17,37,53,101,109,197,257,293,401,409,577,677,701,733,857,1093,1297,
%T 1373,1601,1609,1697,2029,2141,2213,2417,2917,3137,3253,3373,3389,
%U 3853,4261,4357,4493,4909,5209,5477,5641,5801,6257,7057,7229,7573,7937,8101,8837,9029,9413,9613,10009,10429,10453,10613,12101,12109,12553,13457,13693,14177
%N Primes of the form x^2 + y^2 with 0 < x < y such that all the numbers (xa)^2 + (y+a)^2 (a = 1,...,x) are composite.
%C Note that the sequence contains all primes of the form n^2 + 1 with n > 1. A conjecture of Landau states that there are infinitely many primes of the form n^2 + 1.
%C Conjecture: For any prime p > 5 of the form x^2 + y^2 (0 < x < y), there is a prime q not equal to p of the form u^2 + v^2 (0 < u < v) with u + v = x + y.
%C A subsequence of A002313.  _Altug Alkan_, Dec 18 2015
%C Conjecture: each odd number m > 1 is a unique sum m = x + y with 0 < x < y, where x^2 + y^2 is in the sequence.  _Thomas Ordowski_, Jan 16 2017
%H ZhiWei Sun, <a href="/A264904/b264904.txt">Table of n, a(n) for n = 1..3500</a>
%e a(1) = 5 since 5 = 1^2 + 2^2 is a prime with 0 < 1 < 2, and 0^2 + 3^2 = 9 is composite.
%e a(4) = 53 since 53 = 2^2 + 7^2 is a prime with 0 < 2 < 7, and 0^2 + 9^2 = 81 and 1^2 + 8^2 = 65 are both composite.
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
%t Y[n_]:=Y[n]=Sum[If[SQ[n4*y^2],2y,0],{y,0,Sqrt[n/4]}]
%t X[n_]:=X[n]=Sqrt[nY[n]^2]
%t p[n_]:=p[n]=Prime[n]
%t x[n_]:=x[n]=X[p[n]]
%t y[n_]:=y[n]=Y[p[n]]
%t n=0;Do[If[Mod[p[k]1,4]==0,Do[If[PrimeQ[a^2+(x[k]+y[k]a)^2],Goto[aa]],{a,0,Min[x[k],y[k]]1}];n=n+1;Print[n," ",p[k]]];Label[aa];Continue,{k,2,1669}]
%Y Cf. A000040, A000290, A002144, A002313, A002496, A264865, A264866.
%K nonn
%O 1,1
%A _ZhiWei Sun_, Nov 28 2015
