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

%I A378879 #15 Jun 09 2025 09:03:15
%S A378879 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,
%T A378879 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,
%U A378879 2,2,1,5,0,1,1,3,2,2,1,4,4,1,1,4,0,2
%N A378879 a(n) = number of non-Pythagorean primes in the prime factorization of n (including multiplicities).
%F A378879 From _R. J. Mathar_, Jan 28 2025: (Start)
%F A378879 a(n) + A083025(n) = A001222(n).
%F A378879 a(n) = A007814(n)+A065339(n). (End)
%F A378879 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
%e A378879 a(12) = 3 because 12 = 2*2*3, where 2 (with multiplicity 2) and 3 are non-Pythagorean primes.
%p A378879 A378879 := proc(n)
%p A378879     local a,f ;
%p A378879     a := 0 ;
%p A378879     for f in ifactors(n)[2] do
%p A378879         if op(1, f) mod 4 <> 1 then
%p A378879             a := a+op(2, f) ;
%p A378879         end if;
%p A378879     end do:
%p A378879     a ;
%p A378879 end proc:
%p A378879 seq(A378879(n),n=1..50) ; # _R. J. Mathar_, Jan 27 2025
%t A378879 f[{x_, y_}] := If[Mod[x, 4] == 1, y, -y];
%t A378879 s[n_] := Map[f, FactorInteger[n]];
%t A378879 p[n_] := {Total[Select[s[n], # > 0 &]], -Total[Select[s[n], # < 0 &]]};
%t A378879 p[1] = {0, 0};
%t A378879 t = Table[p[n], {n, 1, 135}]
%t A378879 Map[First, t]   (* A083025 *)
%t A378879 Map[Last, t]   (* A378879 *)
%Y A378879 Cf. A001222, A002144, A002145, A007814, A065339, A083025, A378880.
%K A378879 nonn,easy
%O A378879 1,4
%A A378879 _Clark Kimberling_, Jan 14 2025