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A322885
Number of 3-generated Abelian groups of order n.
3
1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 7, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 5, 1, 2, 2, 4, 1, 1
OFFSET
1,4
COMMENTS
Groups generated by fewer than 3 elements are not excluded. The number of Abelian groups with 3 invariant factors is a(n) - A046951(n).
Sum of the first three columns from A249770 (for n > 1).
Dirichlet convolution of A061704 and A010052. Dirichlet convolution of A046951 and A010057.
LINKS
FORMULA
Multiplicative with a(p^e) = A001399(e).
Dirichlet g.f.: zeta(s) * zeta(2s) * zeta(3s).
Sum_{k=1..n} a(k) ~ Pi^2*Zeta(3)*n/6 + Zeta(1/2)*Zeta(3/2)*sqrt(n) + Zeta(1/3)*Zeta(2/3)*n^(1/3). - Vaclav Kotesovec, Feb 02 2019
MAPLE
f:= proc(n) local t;
mul(round((t[2]+3)^2/12), t=ifactors(n)[2])
end proc:
map(f, [$1..200]); # Robert Israel, May 20 2019
MATHEMATICA
a[n_] := Times @@ (Round[(# + 3)^2/12]& /@ FactorInteger[n][[All, 2]]);
Array[a, 102] (* Jean-François Alcover, Jan 02 2019 *)
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Álvar Ibeas, Dec 29 2018
STATUS
approved