Search: id:A097945
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%I A097945
%S A097945 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,
%T A097945 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,
%U A097945 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
%V A097945 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,
%W A097945 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,
%X A097945 30,0,0,48,-20,-66,0,44,-24,-70,0,-72,36,0,0,60,-24,-78,0,0,40,-82,0
%N A097945 a(n) = mu(n)*phi(n) where mu(n) is the Mobius function (A008683) and 
               phi(n) is the Euler totient function (A000010).
%C A097945 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 square-free, while mu(n) = 0 when n is not square-free. 
               - Frank Adams-Watters (FrankTAW(AT)Netscape.net), May 14 2006
%C A097945 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 square-free, 
               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
%C A097945 Multiplicative with a(p^1) = 1-p, a(p^e) = 0, e > 1. Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu) 
               May 24, 2005.
%C A097945 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
%H A097945 <a href="http://en.wikipedia.org/wiki/Euler's_phi_function">Euler's totient 
               function</a> at Wikipedia.org
%t A097945 Table[ MoebiusMu[n]EulerPhi[n], {n, 85}] (from Robert G. Wilson v Sep 
               06 2004)
%Y A097945 Cf. A000010, A008683, A047994.
%Y A097945 Cf. A143153.
%Y A097945 Sequence in context: A094572 A079534 A097042 this_sequence A153733 A083218 
               A139716
%Y A097945 Adjacent sequences: A097942 A097943 A097944 this_sequence A097946 A097947 
               A097948
%K A097945 sign,mult
%O A097945 1,3
%A A097945 Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Sep 04 2004
%E A097945 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Sep 06 2004
%E A097945 Edited by N. J. A. Sloane (njas(AT)research.att.com), May 20 2006

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