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 A097945 a(n) = mu(n)*phi(n) where mu(n) is the Mobius function (A008683) and phi(n) is the Euler totient function (A000010). 24
 1, -1, -2, 0, -4, 2, -6, 0, 0, 4, -10, 0, -12, 6, 8, 0, -16, 0, -18, 0, 12, 10, -22, 0, 0, 12, 0, 0, -28, -8, -30, 0, 20, 16, 24, 0, -36, 18, 24, 0, -40, -12, -42, 0, 0, 22, -46, 0, 0, 0, 32, 0, -52, 0, 40, 0, 36, 28, -58, 0, -60, 30, 0, 0, 48, -20, -66, 0, 44, -24, -70, 0, -72, 36, 0, 0, 60, -24, -78, 0, 0, 40, -82, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Also, a(n) = mu(n)*uphi(n) where mu(n) is the Mobius function (A008683) and uphi(n) is the unitary totient function (A047994), since phi(n) = uphi(n) when n is squarefree, while mu(n) = 0 when n is not squarefree. - Franklin T. Adams-Watters, May 14 2006 Conjecture: Sum_{n>=1} mu(n)/phi(n) = Sum_{n>=1} a(n)/phi(n)^2 = 0. It is true that Sum_{n>=1} mu(n)/phi(n)^s = 0 at least for s > 1 since: phi(2)=1, phi is multiplicative, so for n's that are squarefree, the phi(n) values can be partitioned in pairs where phi(m)=phi(2m) and mu(m) = -mu(2m). So Sum_{i=1..n} mu(i)/phi(i)^s < Sum_{j=floor(n/2)..n} 1/phi(j)^s, which approaches 0 as n increases since (1) n^(1-e) < phi(n) < n for any e > 0 and n > N(e) and (2) Sum_{i..n} 1/n^s converges for s > 1. Conjecture: Sum_{n>=1} mu(n)/phi(n)^z = 0 for Re(z) > 1. Multiplicative with a(p^1) = 1-p, a(p^e) = 0, e > 1. - Mitch Harris, May 24 2005 Row sums of triangle A143153 = a signed version of the sequence such that parity = (-) iff A008683(n) = (+); 0 or (+): (1, 1, 2, 0, 4, -2, 6, 0, 0, -4, 10, 0, 12, -6, 0, 0, 0, ...). - Gary W. Adamson, Jul 27 2008 Dirichlet inverse of A003958. - R. J. Mathar, Jul 08 2011 LINKS Alois P. Heinz, Table of n, a(n) for n = 1..10000 Wikipedia, Euler's totient function FORMULA Dirichlet g.f.: Product_{primes p} (1-p^(1-s)+p^(-s)). - R. J. Mathar, Aug 29 2011 Sum_{d|n} abs(a(d)) = rad(n) = A007947(n). - Rémy Sigrist, Nov 05 2017 Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = A065464/2 = (1/2) * Product_{primes p} (1 - 2/p^2 + 1/p^3) = 0.21412475283854722... Equivalently, c = A065463 * 3 / Pi^2. - Vaclav Kotesovec, Jun 14 2020 From Antti Karttunen, Aug 20 2021: (Start) a(n) = mu(n)*A000010(n) = mu(n)*A003958(n) = mu(n)*A047994(n) = mu(n)*A173557(n), where mu is Möbius mu function (A008683). a(n) = A008966(n) * A023900(n) = abs(mu(n)) * A023900(n). a(n) = A322581(n) - A003958(n). (End) MAPLE with(numtheory): a:= n-> mobius(n)*phi(n): seq(a(n), n=1..100);  # Alois P. Heinz, Aug 06 2012 MATHEMATICA Table[ MoebiusMu[n]EulerPhi[n], {n, 85}] (* Robert G. Wilson v, Sep 06 2004 *) PROG (PARI) a(n)=moebius(n)*eulerphi(n) \\ Charles R Greathouse IV, Feb 21 2013 (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - p*X + X))[n], ", ")) \\ Vaclav Kotesovec, Jun 14 2020 CROSSREFS Cf. A000010, A003958, A007947, A008683, A008966, A023900, A047994, A143153, A173557, A322581. Sequence in context: A337697 A328599 A222303 * A319997 A153733 A083218 Adjacent sequences:  A097942 A097943 A097944 * A097946 A097947 A097948 KEYWORD sign,mult AUTHOR Gerald McGarvey, Sep 04 2004 EXTENSIONS More terms from Robert G. Wilson v, Sep 06 2004 Edited by N. J. A. Sloane, May 20 2006 STATUS approved

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Last modified August 8 10:04 EDT 2022. Contains 356009 sequences. (Running on oeis4.)