A158210 a(n) = omega(n) * (-1)^mu(n), where mu is the Moebius function.

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

Offset: 1

Author

Daniel Forgues, Mar 14 2009

Keywords

• sign

Comments

Numbers k such that: a(k) < -1: A120944; a(k) = -1: A000040, a(k) > -1: A162966; a(k) = +1: A246547; a(k) > +1: A126706.

Links

• Daniel Forgues, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001221 (omega), A008683 (mu).

Cf. A120944, A000040, A162966, A246547, A126706, A013661.

Programs

Mathematica

Table[(-1)^MoebiusMu[n]*PrimeNu[n], {n, 81}] (* L. Edson Jeffery, Dec 08 2014 *)

PARI

a(n) = {my(f= factor(n)); omega(f) * (-1)^moebius(f);} \\ Amiram Eldar, Oct 05 2024

Formula

a(n) = omega(n) * (-1)^mu(n), where mu is the Moebius function.

a(n) = A001221(n) * (-1)^A008683(n).

a(mn) = [|a(m)| + |a(n)|] * max(sign[a(n)], sign[a(m)]), gcd(m,n) = 1, m > 1, n > 1.

Sum_{k=1..n} a(k) = (1-2/zeta(2)) * n * log(log(n)) + O(n). - Amiram Eldar, Oct 05 2024

Extensions

Edited by Joerg Arndt, Feb 12 2024