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%I #22 Aug 31 2023 05:30:42
%S 1,-3,-2,1,-4,6,-6,1,-2,12,-10,-2,-12,18,8,1,-16,6,-18,-4,12,30,-22,
%T -2,-4,36,-2,-6,-28,-24,-30,1,20,48,24,-2,-36,54,24,-4,-40,-36,-42,
%U -10,8,66,-46,-2,-6,12,32,-12,-52,6,40,-6,36,84,-58,8,-60,90,12,1,48,-60,-66,-16,44,-72,-70,-2,-72,108,8,-18
%N Sum_{d|n} a(d)/d = (-1)^(n-1)/n for n>=1; equals the logarithmic g.f. of A117210.
%H Paul D. Hanna, <a href="/A117212/b117212.txt">Table of n, a(n) for n = 1..1000</a>
%F G.f.: Sum_{n>=1} a(n)*x^n/n = log(F(x)), where F(x) is the g.f. of A117210 and satisfies: (1+x) = Product_{n>=1} F(x^n).
%F a(n) = A023900(n) if n (mod 4) = 1 or 3, a(n) = 3*A023900(n) if n (mod 4) = 2, a(n) = -A023900(n) if n (mod 4) = 0, where A023900 is the Dirichlet inverse of Euler totient function.
%F From _Stuart Clary_, Apr 15 2006: (Start)
%F G.f.: A(x) = sum_{k>=1} mu(k) k x^k/(1 + x^k) where mu(k) is the Möbius function, A008683.
%F G.f.: A(x) is x times the logarithmic derivative of A117210(x).
%F G.f.: A(x) = A023900(x) - 2 A023900(x^2).
%F a(n) = sum_{d|n} (-1)^(n/d - 1) mu(d) d.
%F (End)
%F Dirichlet convolution of A055615 and A062157, so the Dirichlet g.f. is the product zeta(s)*(1-2^(1-s))/zeta(s-1). - _R. J. Mathar_, Feb 07 2011
%F Multiplicative with a(2) = -3, a(2^e) = 1 for e >= 2, and a(p^e) = 1 - p for an odd prime p. - _Amiram Eldar_, Aug 31 2023
%e For n=6, Sum_{d|6} a(d)/d = a(1)/1 + a(2)/2 + a(3)/3 + a(6)/6 = 1/1 - 3/2 - 2/3 + 6/6 = -1/6.
%t nmax = 72; Drop[ CoefficientList[ Series[ Sum[ MoebiusMu[k] k x^k/(1 + x^k), {k, 1, nmax} ], {x, 0, nmax} ], x ], 1 ] (* _Stuart Clary_, Apr 15 2006 *)
%t f[p_, e_] := 1 - p; f[2, e_] := If[e == 1, -3, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Aug 31 2023 *)
%o (PARI) a(n)=sumdiv(n,d,d*moebius(d))*[1,3,1,-1][(n-1)%4+1]
%Y Cf. A008683, A023900, A055615, A062157, A117210, A117211.
%K sign,easy,mult
%O 1,2
%A _Paul D. Hanna_, Mar 03 2006
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