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A322885 Number of 3-generated Abelian groups of order n. 2

%I

%S 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,

%T 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,

%U 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

%N Number of 3-generated Abelian groups of order n.

%C Groups generated by fewer than 3 elements are not excluded. The number of Abelian groups with 3 invariant factors is a(n) - A046951(n).

%C Sum of the first three columns from A249770 (for n > 1).

%C Dirichlet convolution of A061704 and A010052. Dirichlet convolution of A046951 and A010057.

%H Robert Israel, <a href="/A322885/b322885.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(p^e) = A001399(e).

%F Dirichlet g.f.: zeta(s) * zeta(2s) * zeta(3s).

%F 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

%p f:= proc(n) local t;

%p mul(round((t[2]+3)^2/12),t=ifactors(n)[2])

%p end proc:

%p map(f, [$1..200]); # _Robert Israel_, May 20 2019

%t a[n_] := Times @@ (Round[(# + 3)^2/12]& /@ FactorInteger[n][[All, 2]]);

%t Array[a, 102] (* _Jean-François Alcover_, Jan 02 2019 *)

%Y Cf. A001399, A010052, A010057, A046951, A061704, A249770.

%K nonn,mult

%O 1,4

%A _Álvar Ibeas_, Dec 29 2018

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Last modified May 28 01:48 EDT 2020. Contains 334671 sequences. (Running on oeis4.)