A118209 Expansion of Sum_{k>=1} lambda(k) * k * x^k/(1 + x^k) where lambda(k) is the Liouville function, A008836.

1, -3, -2, 5, -4, 6, -6, -11, 7, 12, -10, -10, -12, 18, 8, 21, -16, -21, -18, -20, 12, 30, -22, 22, 21, 36, -20, -30, -28, -24, -30, -43, 20, 48, 24, 35, -36, 54, 24, 44, -40, -36, -42, -50, -28, 66, -46, -42, 43, -63, 32, -60, -52, 60, 40, 66, 36, 84, -58, 40, -60, 90, -42, 85, 48, -60, -66, -80, 44, -72, -70, -77, -72, 108, -42

Offset: 1

Author

Stuart Clary, Apr 15 2006

Keywords

• sign
• easy
• mult

Comments

Related to the logarithmic derivative of A118207(x) and A118208(x).

Related to a signed variant of A022998 via Mobius inversion. - R. J. Mathar, Jul 03 2011

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A008836, A022998, A028260, A061019, A061020, A062157, A117212, A118207, A118208.

Programs

Mathematica

nmax = 80; lambda[k_Integer?Positive] := If[ k > 1, (-1)^Total[ Part[Transpose[FactorInteger[k]], 2] ], 1 ]; Drop[ CoefficientList[ Series[ Sum[ lambda[k] k x^k/(1 + x^k), {k, 1, nmax} ], {x, 0, nmax} ], x ], 1 ]
f[p_, e_] := (p*(-p)^e+1)/(p+1); f[2, e_] := ((-1)^e*2^(e+2) - 1)/3; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Aug 12 2023 *)

PARI

a(n) = sumdiv(n, d, (-1)^(n/d - 1)*(-1)^vecsum(factor(d)[,2])*d) \\ Michel Marcus, Dec 10 2016

Formula

a(n) = Sum_{d|n} (-1)^(n/d - 1)*lambda(d)*d, Dirichlet convolution of A061019 and A062157.

G.f.: A(x) is x times the logarithmic derivative of A118207(x).

G.f.: A(x) = A061020(x) - 2 A061020(x^2).

Dirichlet g.f.: zeta(s)*zeta(2s-2)*(1-2^(1-s))/zeta(s-1). - R. J. Mathar, Jul 03 2011

a(n) > 0 for n in A028260. - Michel Marcus, Dec 10 2016

Multiplicative with a(2^e) = ((-1)^e*2^(e+2) - 1)/3, and a(p^e) = (p*(-p)^e+1)/(p+1) for an odd prime p. - Amiram Eldar, Aug 12 2023