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A035109
Numerators in expansion of a certain Dirichlet series.
1
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
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
KEYWORD
nonn,easy
STATUS
approved