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A264904 Primes of the form x^2 + y^2 with 0 < x < y such that all the numbers (x-a)^2 + (y+a)^2 (a = 1,...,x) are composite. 2

%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 (x-a)^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 Zhi-Wei 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[n-4*y^2],2y,0],{y,0,Sqrt[n/4]}]

%t X[n_]:=X[n]=Sqrt[n-Y[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 _Zhi-Wei Sun_, Nov 28 2015

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Last modified April 7 18:19 EDT 2020. Contains 333306 sequences. (Running on oeis4.)