A057653 Odd numbers of form x^2 + y^2.

1, 5, 9, 13, 17, 25, 29, 37, 41, 45, 49, 53, 61, 65, 73, 81, 85, 89, 97, 101, 109, 113, 117, 121, 125, 137, 145, 149, 153, 157, 169, 173, 181, 185, 193, 197, 205, 221, 225, 229, 233, 241, 245, 257, 261, 265, 269, 277, 281, 289, 293, 305, 313, 317, 325, 333, 337

Offset: 1

Author

N. J. A. Sloane, Oct 15 2000

Keywords

• nonn

Comments

Numbers with only odd prime factors and such that all prime factors congruent to 3 modulo 4 occur to an even exponent. - Jean-Christophe Hervé, Oct 24 2015

Odd terms of A020668. - Altug Alkan, Nov 19 2015

Also one half of the numbers that are the sum of two odd squares (without multiplicity). See A097269 for twice the numbers. - Wolfdieter Lang, Jan 12 2017

Links

• Jean-Christophe Hervé, Table of n, a(n) for n = 1..4000
• Joerg Arndt, Plane-filling curves on all uniform grids, arXiv preprint arXiv:1607.02433 [math.CO], 2016.
• J. H. Conway, E. M. Rains and N. J. A. Sloane, On the existence of similar sublattices, Canad. J. Math. 51 (1999), 1300-1306 (Abstract, pdf, ps).

Crossrefs

Odd members of A001481.

Odd members of A020668.

Complement of A084109 in 4k+1 numbers (A016813).

Cf. A016754 (odd squares), A097269.

Programs

Maple

readlib(issqr): for n from 1 to 1001 by 2 do for k from 0 to floor(sqrt(n)) do if issqr(n-k^2) then printf(`%d,`,n); break fi; od:od:

Mathematica

fQ[n_] := Length@ Catch@ Do[If[IntegerQ@ Sqrt[n - k^2], Throw[{k, Sqrt[n - k^2]}], Nothing], {k, Floor[Sqrt@ n]^2}] != 0; Select[Range[1, 340, 2], fQ] (* Michael De Vlieger, Nov 13 2015 *)

PARI

isok(n)=my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]%2 && f[i, 1]%4==3, return(0))); 1;
for(n=1, 1e3, if(isok(n) && n%2==1, print1(n", "))) \\ Altug Alkan, Nov 13 2015

PARI

for(n=0, 1e3, if(if( n<1, n==0, 2 * qfrep([ 1, 0; 0, 4], n)[n]) != 0 && n%2==1, print1(n, ", "))) \\ Altug Alkan, Nov 19 2015

Python

from itertools import count, islice
from sympy import factorint
def A057653_gen(): # generator of terms
    return filter(lambda n:all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(n).items()),count(1,2))
A057653_list = list(islice(A057653_gen(),30)) # Chai Wah Wu, Jun 28 2022

Formula

n = odd square * {product of distinct primes == 1 (mod 4)}.

a(n) = A097269(n)/2. - Wolfdieter Lang, Jan 12 2017

Extensions

More terms from James Sellers, Oct 16 2000