A181104 Dirichlet inverse of Ramanujan's L-series (A000594).

1, 24, -252, 2048, -4830, -6048, 16744, 0, 177147, -115920, -534612, -516096, 577738, 401856, 1217160, 0, 6905934, 4251528, -10661420, -9891840, -4219488, -12830688, -18643272, 0, 48828125, 13865712, 0, 34291712, -128406630, 29211840

Offset

1,2

Comments

Although it is conjectured that A000594(n) is never 0 here a(n)=0 for infinitely many n. Namely a(n)=0 iff n is not cubefree (n is in A046099).

Multiplicative because A000594 is. - Andrew Howroyd, Aug 05 2018

References

B. Cloitre, On the order of absolute convergence of Dirichlet series and the Grand Riemann hypothesis, in preparation 2010-2011 (unpublished as of August 2018).

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Formula

For Re(s)>13/2 we have sum_{n>0}a(n)/n^s*sum_{n>0}A000594(n)/n^s=1. If n is squarefree then a(n)=(-1)^omega(n)*A000594(n).

Mathematica

a[1] = 1; a[n_] := a[n] = -Sum[a[d]*RamanujanTau[n/d], {d, Most[Divisors[n]]}]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Jun 18 2013 *)

Prog

(PARI) a(n)=if(n<2, 1/A000594(1), -1/A000594(1)*sumdiv(n, d, if(n-d, a(d)*A000594(n/d), 0)))
(PARI) seq(n)={dirdiv(vector(n, n, n==1), vector(n, n, ramanujantau(n)))} \\ Andrew Howroyd, Aug 05 2018

Crossrefs

Cf. A000594, A046099

Sequence in context: A000594 A278577 A022716 * A051828 A076847 A296575

Adjacent sequences: A181101 A181102 A181103 * A181105 A181106 A181107

Keyword

sign,mult

Author

Benoit Cloitre, Oct 03 2010

Status

approved