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%I #64 May 27 2023 11:48:41
%S 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,
%T 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,
%U 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
%N a(n) = mu(rad(n)), where mu is the Moebius-function (A008683) and rad is the radical or squarefree kernel (A007947).
%C 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
%C 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
%H Reinhard Zumkeller, <a href="/A076479/b076479.txt">Table of n, a(n) for n = 1..10000</a>
%H Eckford Cohen, <a href="https://doi.org/10.1007/BF01180473">Arithmetical functions associated with the unitary divisors of an integer</a>, Math. Zeitschr., Vol. 74 (1960), pp. 66-80.
%H Jan van de Lune and Robert E. Dressler, <a href="https://doi.org/10.1515/crll.1975.277.117">Some theorems concerning the number theoretic function omega(n)</a>, Journal für die reine und angewandte Mathematik, Vol. 277 (1975), pp. 117-119; <a href="https://eudml.org/doc/151630">alternative link</a>.
%F a(n) = A008683(A007947(n)).
%F a(n) = (-1)^A001221(n). Multiplicative with a(p^e) = -1. - _Vladeta Jovovic_, Dec 03 2002
%F a(n) = sign(A180403(n)). - _Mats Granvik_, Oct 08 2010
%F Sum_{n>=1} a(n)*phi(n)/n^3 = A065463 with phi()=A000010() [Cohen, Lemma 3.5]. - _R. J. Mathar_, Apr 11 2011
%F Dirichlet convolution of A000012 with A226177 (signed variant of A074823 with one factor mu(n) removed). - _R. J. Mathar_, Apr 19 2011
%F Sum_{n>=1} a(n)/n^2 = A065469. - _R. J. Mathar_, Apr 19 2011
%F a(n) = Sum_{d|n} mu(d)*tau_2(d) = Sum_{d|n} A008683(d)*A000005(d) . - _Enrique Pérez Herrero_, Jan 17 2013
%F a(A030230(n)) = -1; a(A030231(n)) = +1. - _Reinhard Zumkeller_, Jun 01 2013
%F Dirichlet g.f.: zeta(s) * Product_{p prime} (1 - 2p^(-s)). - _Álvar Ibeas_, Dec 30 2018
%F Sum_{n>=1} a(n)/n = 0 (van de Lune and Dressler, 1975). - _Amiram Eldar_, Mar 05 2021
%F From _Richard L. Ollerton_, May 07 2021: (Start)
%F For n>1, Sum_{k=1..n} a(gcd(n,k))*phi(gcd(n,k))*rad(gcd(n,k))/gcd(n,k) = 0.
%F 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)
%F a(n) = Sum_{d1|n} Sum_{d2|n} mu(d1*d2). - _Ridouane Oudra_, May 25 2023
%p A076479 := proc(n)
%p (-1)^A001221(n) ;
%p end proc:
%p seq(A076479(n),n=1..80) ; # _R. J. Mathar_, Nov 02 2016
%t Table[(-1)^PrimeNu[n], {n,50}] (* _Enrique Pérez Herrero_, Jan 17 2013 *)
%o (PARI)
%o N=66;
%o mu=vector(N); mu[1]=1;
%o { for (n=2,N,
%o s = 0;
%o fordiv (n,d,
%o if (gcd(d,n/d)!=1, next() ); /* unitary divisors only */
%o s += mu[d];
%o );
%o mu[n] = -s;
%o ); };
%o mu /* _Joerg Arndt_, May 13 2011 */
%o /* omitting the line if ( gcd(...)) gives the usual Moebius function */
%o (PARI) a(n)=(-1)^omega(n) \\ _Charles R Greathouse IV_, Aug 02 2013
%o (Haskell)
%o a076479 = a008683 . a007947 -- _Reinhard Zumkeller_, Jun 01 2013
%o (Magma) [(-1)^(#PrimeDivisors(n)): n in [1..100]]; // _Vincenzo Librandi_, Dec 31 2018
%o (Python)
%o from math import prod
%o from sympy.ntheory import mobius, primefactors
%o def A076479(n): return mobius(prod(primefactors(n))) # _Chai Wah Wu_, Sep 24 2021
%Y Cf. A000005, A000010, A001221, A007947, A008683, A008836, A030230, A065469, A076480, A180403, A226177.
%Y Cf. A174863 (partial sums).
%K sign,mult
%O 1,1
%A _Reinhard Zumkeller_, Oct 14 2002
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