A035109 Numerators in expansion of a certain Dirichlet series.

1, 1, 5, 1, 7, 5, 9, 1, 17, 7, 13, 5, 15, 9, 35, 1, 19, 17, 21, 7, 45, 13, 25, 5, 37, 15, 53, 9, 31, 35, 33, 1, 65, 19, 63, 17, 39, 21, 75, 7, 43, 45, 45, 13, 119, 25, 49, 5, 65, 37, 95, 15, 55, 53, 91, 9, 105, 31, 61, 35, 63, 33, 153, 1, 105, 65, 69, 19, 125, 63, 73, 17, 75, 39

Offset

0,3

Comments

a(n) is also the number of orbits of length n for the map SxT where S has one orbit of each length and T has one orbit of each odd length. - Thomas Ward, Apr 08 2009

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

M. Baake and R. V. Moody, Similarity submodules and semigroups in Quasicrystals and Discrete Geometry, ed. J. Patera, Fields Institute Monographs, vol. 10 AMS, Providence, RI (1998) pp. 1-13.

A. Pakapongpun, T. Ward, Functorial Orbit counting, JIS 12 (2009) 09.2.4, example 17.

Formula

Dirichlet g.f.: zeta(s)*Product((1+p^-s)/(1-p^(1-s))), p > 2.

a(n) = (1/n)*sumdiv(n,d,mu(n/d)sum(d,e,e)sum(d,e odd only,e). - Thomas Ward, Apr 08 2009

Example

a(6) = (1/6)*(mu(6)*1*1 + mu(3)*3*1 + mu(2)*4*4 + mu(1)*4*12) = 5. - Thomas Ward, Apr 08 2009

Mathematica

a[n_] := (1/n)*DivisorSum[n, MoebiusMu[n/#]*DivisorSigma[1, #]*DivisorSum[ #, If[OddQ[#], #, 0]&]&]; Array[a, 80] (* Jean-François Alcover, Dec 07 2015, adapted from PARI *)

Prog

(PARI) a(n)=(1/n)*sumdiv(n, d, moebius(n/d)*sigma(d)*sumdiv(d, e, if(e%2, e, 0))) \\ Thomas Ward, Apr 08 2009

Crossrefs

Sequence in context: A198129 A244425 A332343 * A301509 A101263 A187561

Adjacent sequences: A035106 A035107 A035108 * A035110 A035111 A035112

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved