

A264904


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.


2



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
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OFFSET

1,1


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


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..3500


EXAMPLE

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.


MATHEMATICA

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Y[n_]:=Y[n]=Sum[If[SQ[n4*y^2], 2y, 0], {y, 0, Sqrt[n/4]}]
X[n_]:=X[n]=Sqrt[nY[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}]


CROSSREFS

Cf. A000040, A000290, A002144, A002313, A002496, A264865, A264866.
Sequence in context: A080167 A060245 A119456 * A257582 A273538 A273212
Adjacent sequences: A264901 A264902 A264903 * A264905 A264906 A264907


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Nov 28 2015


STATUS

approved



