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A076479 a(n) = mu(rad(n)), where mu is the Moebius-function (A008683) and rad is the radical or squarefree kernel (A007947). 25
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 (list; graph; refs; listen; history; text; internal format)
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. 74 (1960) 66-80.

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). [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

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

CROSSREFS

Cf. A076480, A008836, A226177.

Cf. A174863 (partial sums).

Sequence in context: A143622 A246016 A306638 * A155040 A209661 A033999

Adjacent sequences:  A076476 A076477 A076478 * A076480 A076481 A076482

KEYWORD

sign,mult

AUTHOR

Reinhard Zumkeller, Oct 14 2002

STATUS

approved

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Last modified October 22 00:52 EDT 2019. Contains 328315 sequences. (Running on oeis4.)