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A076479
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mu(sfk(n)), where mu is the Moebius-function (A008683) and sfk is the squarefree kernel (A007947).
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8
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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
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OFFSET
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1,1
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COMMENTS
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a(n)=A008683(A007947(n)).
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. [From 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). [From Laszlo Toth, Oct 08 2009]
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REFERENCES
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Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Math. Zeitschr. 74 (1960) 66-80
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LINKS
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Table of n, a(n) for n=1..87.
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FORMULA
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a(n) = (-1)^A001221(n). Multiplicative with a(p^e) = -1. - Vladeta Jovovic, Dec 03 2002
a(n)=sign(A180403). [From 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 the signed variant of A074823, i.e., the variant where one factor mu(n) is removed from the definition of A074823. - 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
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MATHEMATICA
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Table[(-1)^PrimeNu[n], {n, 50}] (* Enrique Pérez Herrero, Jan 17 2013 *)
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PROG
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(Pari)
N=66; /* that many terms */
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 /* show terms */ /* Joerg Arndt, May 13 2011 */
/* omitting the line if ( gcd(...)) gives the usual Moebius function */
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CROSSREFS
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Cf. A076480.
Cf. A008836.
Sequence in context: A098417 A143622 * A155040 A033999 A000012 A162511
Adjacent sequences: A076476 A076477 A076478 * A076480 A076481 A076482
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KEYWORD
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sign,mult,changed
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AUTHOR
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Reinhard Zumkeller, Oct 14 2002
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STATUS
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approved
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