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.

5, 17, 37, 53, 101, 109, 197, 257, 293, 401, 409, 577, 677, 701, 733, 857, 1093, 1297, 1373, 1601, 1609, 1697, 2029, 2141, 2213, 2417, 2917, 3137, 3253, 3373, 3389, 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

Offset: 1

Author

Zhi-Wei Sun, Nov 28 2015

Keywords

• nonn

Comments

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.

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.

A subsequence of A002313. - Altug Alkan, Dec 18 2015

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

Examples

			a(1) = 5 since 5 = 1^2 + 2^2 is a prime with 0 < 1 < 2, and 0^2 + 3^2 = 9 is composite.
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.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..3500

Crossrefs

Cf. A000040, A000290, A002144, A002313, A002496, A264865, A264866.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Y[n_]:=Y[n]=Sum[If[SQ[n-4*y^2],2y,0],{y,0,Sqrt[n/4]}]
X[n_]:=X[n]=Sqrt[n-Y[n]^2]
p[n_]:=p[n]=Prime[n]
x[n_]:=x[n]=X[p[n]]
y[n_]:=y[n]=Y[p[n]]
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}]