A316523 Number of odd multiplicities minus number of even multiplicities in the canonical prime factorization of n.

0, 1, 1, -1, 1, 2, 1, 1, -1, 2, 1, 0, 1, 2, 2, -1, 1, 0, 1, 0, 2, 2, 1, 2, -1, 2, 1, 0, 1, 3, 1, 1, 2, 2, 2, -2, 1, 2, 2, 2, 1, 3, 1, 0, 0, 2, 1, 0, -1, 0, 2, 0, 1, 2, 2, 2, 2, 2, 1, 1, 1, 2, 0, -1, 2, 3, 1, 0, 2, 3, 1, 0, 1, 2, 0, 0, 2, 3, 1, 0, -1, 2, 1, 1

Offset: 1

Author

Gus Wiseman, Jul 05 2018

Keywords

• sign
• easy

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A000607, A071321, A100118, A112798.

Cf. A187039 (where a(n)=0). - Michel Marcus, Jul 08 2018

Cf. A077761, A162641, A162642, A179119.

Programs

Maple

f:= proc(n) local F;
  F:= map(t -> t[2],ifactors(n)[2]);
  2*nops(select(type,F,odd))-nops(F);
end proc:
map(f, [$1..100]); # Robert Israel, Aug 27 2018

Mathematica

Table[Total[-(-1)^If[n==1,{},FactorInteger[n][[All,2]]]],{n,100}]

PARI

a(n) = my(f=factor(n)); -sum(k=1, #f~, (-1)^(f[k,2])); \\ Michel Marcus, Jul 08 2018; corrected Jun 13 2022

Formula

If i and j are coprime, a(i*j) = a(i)+a(j). - Robert Israel, Aug 27 2018

From Amiram Eldar, Oct 05 2023: (Start)

Additive with a(p^e) = (-1)^(e+1).

a(n) = A162642(n) - A162641(n).

Sum_{k=1..n} a(k) = n * log(log(n)) + c * n + O(n/log(n)), where c = A077761 - 2*A179119 = -0.398962... . (End)