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 and 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) * Sum_{d|n} mu(n/d) * (Sum_{e|d} e) * (Sum_{e|d, e odd only} e). - Thomas Ward, Apr 08 2009
From Ridouane Oudra, Jun 18 2025: (Start)
a(n) = Sum_{d|n} (psi(2*d) - 2*psi(d)), where psi = A001615.
a(n) = Sum_{d|n, d odd} psi(d).
a(n) = A309324(n) / gcd(n,2).
a(2*n) = a(n).
a(2*n+1) = A060648(2*n+1). (End)
From Vaclav Kotesovec, Jun 21 2025: (Start)
Dirichlet g.f.: (1 - 2^(1-s)) * zeta(s-1) * zeta(s)^2 / ((1 + 2^(-s)) * zeta(2*s)).
Sum_{k=1..n} a(k) ~ n^2/2. (End)
EXAMPLE
a(6) = (1/6)*(mu(6)*1*1 + mu(3)*3*1 + mu(2)*4*4 + mu(1)*12*4) = 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
KEYWORD
nonn,easy
AUTHOR
STATUS
approved
