A263715 Nonnegative integers that are the sum or difference of two squares.

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 39, 40, 41, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 79, 80

Offset: 1

Author

Jean-Christophe Hervé, Oct 24 2015

Keywords

• nonn

Comments

Contains all integers that are not equal to 2 (mod 4) (they are of the form y^2 - x^2) and those of the form 4k+2 = 2*(2k+1) with the odd number 2k+1 equal to the sum of two squares (A057653).

Examples

			2 = 1^2 + 1^2, 3 = 2^2 - 1^2, 4 = 2^2 + 0^2, 5 = 2^2 + 1^2 = 3^2 - 2^2.
		

Links

• Jean-Christophe Hervé, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001481, A057653, A097269, A042965, A020668, A263737.

Cf. A062316 (complement), A079298.

Programs

Mathematica

r[n_] := Reduce[n == x^2 + y^2, {x, y}, Integers] || Reduce[0 <= y <= x && n == x^2 - y^2, {x, y}, Integers]; Reap[Do[If[r[n] =!= False, Sow[n]], {n, 0, 80}]][[2, 1]] (* Jean-François Alcover, Oct 25 2015 *)

Python

from itertools import count, islice
from sympy import factorint
def A263715_gen(): # generator of terms
    return filter(lambda n: n & 3 != 2 or all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(n).items()),count(0))
A263715_list = list(islice(A263715_gen(),30)) # Chai Wah Wu, Jun 28 2022

Formula

Union of A001481 (sums of two squares) and A042965 (differences of two squares).

Union of A042965 and 2*A057653 = A097269, with intersection of A042965 and A097269 = {}.

Union of A020668 (x^2+y^2 and a^2-b^2), A097269 (x^2+y^2, not a^2-b^2) and A263737 (not x^2+y^2, a^2-b^2).