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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).

%I A076479 #69 Jan 11 2025 05:20:51
%S A076479 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 A076479 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 A076479 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 A076479 a(n) = mu(rad(n)), where mu is the Moebius-function (A008683) and rad is the radical or squarefree kernel (A007947).
%C A076479 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 A076479 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 A076479 Reinhard Zumkeller, <a href="/A076479/b076479.txt">Table of n, a(n) for n = 1..10000</a>
%H A076479 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 A076479 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 A076479 a(n) = A008683(A007947(n)).
%F A076479 a(n) = (-1)^A001221(n). Multiplicative with a(p^e) = -1. - _Vladeta Jovovic_, Dec 03 2002
%F A076479 a(n) = sign(A180403(n)). - _Mats Granvik_, Oct 08 2010
%F A076479 Sum_{n>=1} a(n)*phi(n)/n^3 = A065463 with phi()=A000010() [Cohen, Lemma 3.5]. - _R. J. Mathar_, Apr 11 2011
%F A076479 Dirichlet convolution of A000012 with A226177 (signed variant of A074823 with one factor mu(n) removed). - _R. J. Mathar_, Apr 19 2011
%F A076479 Sum_{n>=1} a(n)/n^2 = A065469. - _R. J. Mathar_, Apr 19 2011
%F A076479 a(n) = Sum_{d|n} mu(d)*tau_2(d) = Sum_{d|n} A008683(d)*A000005(d) . - _Enrique Pérez Herrero_, Jan 17 2013
%F A076479 a(A030230(n)) = -1; a(A030231(n)) = +1. - _Reinhard Zumkeller_, Jun 01 2013
%F A076479 Dirichlet g.f.: zeta(s) * Product_{p prime} (1 - 2p^(-s)). - _Álvar Ibeas_, Dec 30 2018
%F A076479 Sum_{n>=1} a(n)/n = 0 (van de Lune and Dressler, 1975). - _Amiram Eldar_, Mar 05 2021
%F A076479 From _Richard L. Ollerton_, May 07 2021: (Start)
%F A076479 For n>1, Sum_{k=1..n} a(gcd(n,k))*phi(gcd(n,k))*rad(gcd(n,k))/gcd(n,k) = 0.
%F A076479 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 A076479 a(n) = Sum_{d1|n} Sum_{d2|n} mu(d1*d2). - _Ridouane Oudra_, May 25 2023
%p A076479 A076479 := proc(n)
%p A076479     (-1)^A001221(n) ;
%p A076479 end proc:
%p A076479 seq(A076479(n),n=1..80) ; # _R. J. Mathar_, Nov 02 2016
%t A076479 Table[(-1)^PrimeNu[n], {n,50}] (* _Enrique Pérez Herrero_, Jan 17 2013 *)
%o A076479 (PARI)
%o A076479 N=66;
%o A076479 mu=vector(N); mu[1]=1;
%o A076479 { for (n=2,N,
%o A076479     s = 0;
%o A076479     fordiv (n,d,
%o A076479         if (gcd(d,n/d)!=1, next() ); /* unitary divisors only */
%o A076479         s += mu[d];
%o A076479     );
%o A076479     mu[n] = -s;
%o A076479 ); };
%o A076479 mu /* _Joerg Arndt_, May 13 2011 */
%o A076479 /* omitting the line if ( gcd(...)) gives the usual Moebius function */
%o A076479 (PARI) a(n)=(-1)^omega(n) \\ _Charles R Greathouse IV_, Aug 02 2013
%o A076479 (Haskell)
%o A076479 a076479 = a008683 . a007947  -- _Reinhard Zumkeller_, Jun 01 2013
%o A076479 (Magma) [(-1)^(#PrimeDivisors(n)): n in [1..100]]; // _Vincenzo Librandi_, Dec 31 2018
%o A076479 (Python)
%o A076479 from math import prod
%o A076479 from sympy.ntheory import mobius, primefactors
%o A076479 def A076479(n): return mobius(prod(primefactors(n))) # _Chai Wah Wu_, Sep 24 2021
%Y A076479 Cf. A000005, A000010, A001221, A007947, A008683, A008836, A030230, A065469, A076480, A180403, A226177.
%Y A076479 Cf. A174863 (partial sums).
%K A076479 sign,mult,easy
%O A076479 1,1
%A A076479 _Reinhard Zumkeller_, Oct 14 2002