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

1,3

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).

Links

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

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).

Example

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

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 *)

Prog

(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

Crossrefs

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

Cf. A062316 (complement), A079298.

Sequence in context: A088451 A047595 A079298 * A023055 A359528 A365819

Adjacent sequences: A263712 A263713 A263714 * A263716 A263717 A263718

Keyword

nonn

Author

Jean-Christophe Hervé, Oct 24 2015

Status

approved