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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). 32
 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., Vol. 74 (1960), pp. 66-80. Jan van de Lune and Robert E. Dressler, Some theorems concerning the number theoretic function omega(n), Journal für die reine und angewandte Mathematik, Vol. 277 (1975), pp. 117-119; alternative link. 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(n)). - 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 Sum_{n>=1} a(n)/n = 0 (van de Lune and Dressler, 1975). - Amiram Eldar, Mar 05 2021 From Richard L. Ollerton, May 07 2021: (Start) 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) 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. A000005, A000010, A001221, A007947, A008683, A008836, A030230, A065469, A076480, A180403, 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 June 12 06:52 EDT 2021. Contains 344943 sequences. (Running on oeis4.)