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

1,1

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

Zhi-Wei 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[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}]

Crossrefs

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

Sequence in context: A080167 A060245 A119456 * A257582 A339988 A273538

Adjacent sequences: A264901 A264902 A264903 * A264905 A264906 A264907

Keyword

nonn

Author

Zhi-Wei Sun, Nov 28 2015

Status

approved