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A117212
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Sum_{d|n} a(d)/d = (-1)^(n-1)/n for n>=1; equals the logarithmic g.f. of A117210.
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7
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1, -3, -2, 1, -4, 6, -6, 1, -2, 12, -10, -2, -12, 18, 8, 1, -16, 6, -18, -4, 12, 30, -22, -2, -4, 36, -2, -6, -28, -24, -30, 1, 20, 48, 24, -2, -36, 54, 24, -4, -40, -36, -42, -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
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OFFSET
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1,2
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LINKS
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FORMULA
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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).
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.
G.f.: A(x) = sum_{k>=1} mu(k) k x^k/(1 + x^k) where mu(k) is the Möbius function, A008683.
G.f.: A(x) is x times the logarithmic derivative of A117210(x).
a(n) = sum_{d|n} (-1)^(n/d - 1) mu(d) d.
(End)
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
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
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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 *)
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PROG
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(PARI) a(n)=sumdiv(n, d, d*moebius(d))*[1, 3, 1, -1][(n-1)%4+1]
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CROSSREFS
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KEYWORD
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sign,easy,mult
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AUTHOR
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STATUS
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approved
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