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

1,4

Links

Table of n, a(n) for n=1..86.

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

Example

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

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 *)

Crossrefs

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

Sequence in context: A213423 A339374 A265753 * A138110 A080233 A156644

Adjacent sequences: A378876 A378877 A378878 * A378880 A378881 A378882

Keyword

nonn,easy

Author

Clark Kimberling, Jan 14 2025

Status

approved