A378879 a(n) = number of non-Pythagorean primes in the prime factorization of n (including multiplicities).

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

Offset: 1

Author

Clark Kimberling, Jan 14 2025

Keywords

• nonn
• easy

Examples

			a(12) = 3 because 12 = 2*2*3, where 2 (with multiplicity 2) and 3 are non-Pythagorean primes.
		

Crossrefs

Cf. A001222, A002144, A002145, A007814, A065339, A083025, A378880.

Programs

Maple

A378879 := proc(n)
    local a,f ;
    a := 0 ;
    for f in ifactors(n)[2] do
        if op(1, f) mod 4 <> 1 then
            a := a+op(2, f) ;
        end if;
    end do:
    a ;
end proc:
seq(A378879(n),n=1..50) ; # R. J. Mathar, Jan 27 2025

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}]
Map[First, t]   (* A083025 *)
Map[Last, t]   (* A378879 *)

Formula

From R. J. Mathar, Jan 28 2025: (Start)

a(n) + A083025(n) = A001222(n).

a(n) = A007814(n)+A065339(n). (End)

Totally additive with a(p) = 1 if p = 2 or p == 3 (mod 4), and a(p) = 0 if p == 1 (mod 4). - Amiram Eldar, Jun 09 2025