OFFSET
1,1
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.
FORMULA
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
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
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 *)
PROG
(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
(Haskell)
a076479 = a008683 . a007947 -- Reinhard Zumkeller, Jun 01 2013
(Magma) [(-1)^(#PrimeDivisors(n)): n in [1..100]]; // Vincenzo Librandi, Dec 31 2018
(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
CROSSREFS
KEYWORD
sign,mult
AUTHOR
Reinhard Zumkeller, Oct 14 2002
STATUS
approved