A076479 a(n) = mu(rad(n)), where mu is the Moebius-function (A008683) and rad is the radical or squarefree kernel (A007947).

1, -1, -1, -1, -1, 1, -1, -1, -1, 1, -1, 1, -1, 1, 1, -1, -1, 1, -1, 1, 1, 1, -1, 1, -1, 1, -1, 1, -1, -1, -1, -1, 1, 1, 1, 1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, 1, -1, 1, 1, 1, -1, 1, 1, 1, 1, 1, -1, -1, -1, 1, 1, -1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, 1, 1, 1, -1, -1, 1, -1, 1, -1, -1, 1, 1, 1

Offset: 1

Author

Reinhard Zumkeller, Oct 14 2002

Keywords

• sign
• mult
• easy

Comments

Multiplicative: a(1) = 1, a(n) for n >=2 is sign of parity of number of distinct primes dividing n. a(p) = -1, a(pq) = 1, a(pq...z) = (-1)^k, a(p^k) = -1, where p,q,.. z are distinct primes and k natural numbers. - Jaroslav Krizek, Mar 17 2009

a(n) is the unitary Moebius function, i.e., the inverse of the constant 1 function under the unitary convolution defined by (f X g)(n)= sum of f(d)g(n/d), where the sum is over the unitary divisors d of n (divisors d of n such that gcd(d,n/d)=1). - Laszlo Toth, Oct 08 2009

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Math. Zeitschr., Vol. 74 (1960), pp. 66-80.
• Jan van de Lune and Robert E. Dressler, Some theorems concerning the number theoretic function omega(n), Journal für die reine und angewandte Mathematik, Vol. 277 (1975), pp. 117-119; alternative link.

Crossrefs

Cf. A000005, A000010, A001221, A007947, A008683, A008836, A030230, A065469, A076480, A180403, A226177.

Cf. A174863 (partial sums).

Programs

Haskell

a076479 = a008683 . a007947  -- Reinhard Zumkeller, Jun 01 2013

Magma

[(-1)^(#PrimeDivisors(n)): n in [1..100]]; // Vincenzo Librandi, Dec 31 2018

Maple

A076479 := proc(n)
    (-1)^A001221(n) ;
end proc:
seq(A076479(n),n=1..80) ; # R. J. Mathar, Nov 02 2016

Mathematica

Table[(-1)^PrimeNu[n], {n,50}] (* Enrique Pérez Herrero, Jan 17 2013 *)

PARI

N=66;
mu=vector(N); mu[1]=1;
{ for (n=2,N,
    s = 0;
    fordiv (n,d,
        if (gcd(d,n/d)!=1, next() ); /* unitary divisors only */
        s += mu[d];
    );
    mu[n] = -s;
); };
mu /* Joerg Arndt, May 13 2011 */
/* omitting the line if ( gcd(...)) gives the usual Moebius function */

PARI

a(n)=(-1)^omega(n) \\ Charles R Greathouse IV, Aug 02 2013

Python

from math import prod
from sympy.ntheory import mobius, primefactors
def A076479(n): return mobius(prod(primefactors(n))) # Chai Wah Wu, Sep 24 2021

Formula

a(n) = A008683(A007947(n)).

a(n) = (-1)^A001221(n). Multiplicative with a(p^e) = -1. - Vladeta Jovovic, Dec 03 2002

a(n) = sign(A180403(n)). - Mats Granvik, Oct 08 2010

Sum_{n>=1} a(n)*phi(n)/n^3 = A065463 with phi()=A000010() [Cohen, Lemma 3.5]. - R. J. Mathar, Apr 11 2011

Dirichlet convolution of A000012 with A226177 (signed variant of A074823 with one factor mu(n) removed). - R. J. Mathar, Apr 19 2011

Sum_{n>=1} a(n)/n^2 = A065469. - R. J. Mathar, Apr 19 2011

a(n) = Sum_{d|n} mu(d)*tau_2(d) = Sum_{d|n} A008683(d)*A000005(d) . - Enrique Pérez Herrero, Jan 17 2013

a(A030230(n)) = -1; a(A030231(n)) = +1. - Reinhard Zumkeller, Jun 01 2013

Dirichlet g.f.: zeta(s) * Product_{p prime} (1 - 2p^(-s)). - Álvar Ibeas, Dec 30 2018

Sum_{n>=1} a(n)/n = 0 (van de Lune and Dressler, 1975). - Amiram Eldar, Mar 05 2021

From Richard L. Ollerton, May 07 2021: (Start)

For n>1, Sum_{k=1..n} a(gcd(n,k))*phi(gcd(n,k))*rad(gcd(n,k))/gcd(n,k) = 0.

For n>1, Sum_{k=1..n} a(n/gcd(n,k))*phi(gcd(n,k))*rad(n/gcd(n,k))*gcd(n,k) = 0. (End)

a(n) = Sum_{d1|n} Sum_{d2|n} mu(d1*d2). - Ridouane Oudra, May 25 2023