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

%I A322885 #20 Jun 12 2025 10:24:35
%S A322885 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 A322885 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 A322885 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 A322885 Number of 3-generated Abelian groups of order n.
%C A322885 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 A322885 Sum of the first three columns from A249770 (for n > 1).
%C A322885 Dirichlet convolution of A061704 and A010052. Dirichlet convolution of A046951 and A010057.
%C A322885 The number of unordered factorizations of n into biquadratefree power of primes (1 and primes, squares of primes and cubes of primes, A087797). - _Amiram Eldar_, Jun 12 2025
%H A322885 Robert Israel, <a href="/A322885/b322885.txt">Table of n, a(n) for n = 1..10000</a>
%F A322885 Multiplicative with a(p^e) = A001399(e).
%F A322885 Dirichlet g.f.: zeta(s) * zeta(2s) * zeta(3s).
%F A322885 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 A322885 f:= proc(n) local t;
%p A322885   mul(round((t[2]+3)^2/12),t=ifactors(n)[2])
%p A322885 end proc:
%p A322885 map(f, [$1..200]); # _Robert Israel_, May 20 2019
%t A322885 a[n_] := Times @@ (Round[(# + 3)^2/12]& /@ FactorInteger[n][[All, 2]]);
%t A322885 Array[a, 102] (* _Jean-François Alcover_, Jan 02 2019 *)
%o A322885 (PARI) a(n) = vecprod(apply(x -> round((x+3)^2/12), factor(n)[, 2])); \\ _Amiram Eldar_, Jun 12 2025
%Y A322885 Cf. A001399, A010052, A010057, A046951, A061704, A087797, A249770.
%K A322885 nonn,mult,easy
%O A322885 1,4
%A A322885 _Álvar Ibeas_, Dec 29 2018