A263737 Nonnegative integers that are the difference of two squares but not the sum of two squares.

3, 7, 11, 12, 15, 19, 21, 23, 24, 27, 28, 31, 33, 35, 39, 43, 44, 47, 48, 51, 55, 56, 57, 59, 60, 63, 67, 69, 71, 75, 76, 77, 79, 83, 84, 87, 88, 91, 92, 93, 95, 96, 99, 103, 105, 107, 108, 111, 112, 115, 119, 120, 123, 124, 127, 129, 131, 132, 133, 135, 139, 140

Offset: 1

Author

Jean-Christophe Hervé, Oct 25 2015

Keywords

• nonn

Comments

Intersection of A022544 (not the sum of two squares) and A042965 (differences of two squares).

The sequence contains all 4k + 3 and no 4k + 2 integers, and some 4k (4*A022544) and 4k+1 (A084109) integers. First differences are thus 1, 2, 3 or 4, each occurring infinitely often.

Links

• Jean-Christophe Hervé, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A001481, A022544, A042965, A016825, A020668, A062316, A097269, A263715.

Programs

Mathematica

rs[n_] := Reduce[n == x^2 + y^2, {x, y}, Integers]; rd[n_] := Reduce[0 <= y <= x && n == x^2 - y^2, {x, y}, Integers]; Reap[Do[If[rs[n] == False && rd[n] =!= False, Sow[n]], {n, 0, 140}]][[2, 1]] (* Jean-François Alcover, Oct 26 2015 *)

Python

from itertools import count, islice
from sympy import factorint
def A263737_gen(): # generator of terms
    return filter(lambda n:n & 3 != 2 and any(p & 3 == 3 and e & 1 for p, e in factorint(n).items()),count(0))
A263737_list = list(islice(A263737_gen(),30)) # Chai Wah Wu, Jun 28 2022