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A117212 Sum_{d|n} a(d)/d = (-1)^(n-1)/n for n>=1; equals the logarithmic g.f. of A117210. 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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).

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

LINKS

Paul D. Hanna, Table of n, a(n) for n = 1..1000

FORMULA

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.

From Stuart Clary, Apr 15 2006: (Start)

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).

G.f.: A(x) = A023900(x) - 2 A023900(x^2).

a(n) = sum_{d|n} (-1)^(n/d - 1) mu(d) d.

(End)

EXAMPLE

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.

MATHEMATICA

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 *)

PROG

(PARI) a(n)=sumdiv(n, d, d*moebius(d))*[1, 3, 1, -1][(n-1)%4+1]

CROSSREFS

Cf. A023900, A117210, A117211.

Sequence in context: A104509 A271513 A306801 * A208153 A105033 A092486

Adjacent sequences:  A117209 A117210 A117211 * A117213 A117214 A117215

KEYWORD

sign,mult

AUTHOR

Paul D. Hanna, Mar 03 2006

STATUS

approved

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Last modified May 21 07:24 EDT 2019. Contains 323441 sequences. (Running on oeis4.)