A377625 -id:A377625 - cp's OEIS frontend

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

Clark Kimberling, Jan 14 2025

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

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

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 *)

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

A380657 Numbers whose prime factorization has more Pythagorean prime factors than non-Pythagorean prime factors (including multiplicities).

Original entry on oeis.org

5, 13, 17, 25, 29, 37, 41, 50, 53, 61, 65, 73, 75, 85, 89, 97, 101, 109, 113, 125, 130, 137, 145, 149, 157, 169, 170, 173, 175, 181, 185, 193, 195, 197, 205, 221, 229, 233, 241, 250, 255, 257, 265, 269, 275, 277, 281, 289, 290, 293, 305, 313, 317, 325, 337

Offset: 1

Clark Kimberling, Jan 30 2025

nonn

			50 appears because 2*5*5 has 2 Pythagorean prime factors but only 1 non-Pythagorean prime factor.

Cf. A000040, A002144, A002145, A083025, A376961, A377625, A378137, A378139, A378140.

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, 500}];
u = Map[First, t];  (* A083025 *)
v = Map[Last, t] ;  (* A376961 *)
v - u (* A377625 *);
Flatten[Position[v - u, -1]]  (* this sequence *)

cp's OEIS Frontend

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

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