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A076479
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a(n) = mu(rad(n)), where mu is the Moebius-function (A008683) and rad is the radical or squarefree kernel (A007947).
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39
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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
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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FORMULA
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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
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)
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MAPLE
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end proc:
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MATHEMATICA
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PROG
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(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;
); };
/* omitting the line if ( gcd(...)) gives the usual Moebius function */
(Haskell)
(Python)
from math import prod
from sympy.ntheory import mobius, primefactors
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CROSSREFS
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Cf. A000005, A000010, A001221, A007947, A008683, A008836, A030230, A065469, A076480, A180403, A226177.
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KEYWORD
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sign,mult
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AUTHOR
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STATUS
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approved
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