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A097945
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a(n) = mu(n)*phi(n) where mu(n) is the Mobius function (A008683) and phi(n) is the Euler totient function (A000010).
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8
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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; internal format)
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OFFSET
| 1,3
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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. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), May 14 2006
Conjecture: Sum_n=1..inf mu(n)/phi(n) = Sum_n=1..inf a(n)/phi(n)^2 = 0 It is true that Sum_n=1..inf 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=[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..inf 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 (Harris.Mitchell(AT)mgh.harvard.edu) 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 (qntmpkt(AT)yahoo.com), Jul 27 2008
Dirichlet inverse of A003958. - R. J. Mathar, Jul 08 2011
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LINKS
| Euler's totient function at Wikipedia.org
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FORMULA
| Dirichlet g.f. product_{primes p} (1-p^(1-s)+p^(-s)). - R. J. Mathar, Aug 29 2011
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MATHEMATICA
| Table[ MoebiusMu[n]EulerPhi[n], {n, 85}] (from Robert G. Wilson v Sep 06 2004)
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CROSSREFS
| Cf. A000010, A008683, A047994.
Cf. A143153.
Sequence in context: A079534 A097042 A196606 * A153733 A083218 A203908
Adjacent sequences: A097942 A097943 A097944 * A097946 A097947 A097948
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KEYWORD
| sign,mult
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AUTHOR
| Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Sep 04 2004
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EXTENSIONS
| More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Sep 06 2004
Edited by N. J. A. Sloane (njas(AT)research.att.com), May 20 2006
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