A378880 a(n) = A378879(n) - A083025(n).

0, 1, 1, 2, -1, 2, 1, 3, 2, 0, 1, 3, -1, 2, 0, 4, -1, 3, 1, 1, 2, 2, 1, 4, -2, 0, 3, 3, -1, 1, 1, 5, 2, 0, 0, 4, -1, 2, 0, 2, -1, 3, 1, 3, 1, 2, 1, 5, 2, -1, 0, 1, -1, 4, 0, 4, 2, 0, 1, 2, -1, 2, 3, 6, -2, 3, 1, 1, 2, 1, 1, 5, -1, 0, -1, 3, 2, 1, 1, 3, 4, 0

Offset: 1

Author

Clark Kimberling, Jan 14 2025

Keywords

• sign
• easy

Examples

			a(10) = 0 because the factorization 2*5 has 1 each of a Pythagorean prime, 5, and a non-Pythagorean prime, 2.
		

Crossrefs

Cf. A000040, A002144, A002145, A083025, A378879.

Programs

Mathematica

f[{x_, y_}] := If[Mod[x, 4] == 1, y, -y];
s[n_] := Map[f, FactorInteger[n]];
p[n_] := {Total[Select[s[n], # > 0 &]], -Total[Select[s[n], # < 0 &]]};
p[1] = {0, 0};
t = Table[p[n], {n, 1, 135}]
u = Map[First, t]   (* A083025 *)
v = Map[Last, t]   (* A378879 *)
v - u (* A377625 *)

Formula

Totally additive with a(p) = 1 if p = 2 or p == 3 (mod 4), and a(p) = -1 if p == 1 (mod 4). - Amiram Eldar, Jun 09 2025