A249770 -id:A249770 - cp's OEIS frontend

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

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

Álvar Ibeas, Dec 29 2018

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.
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

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001399, A010052, A010057, A046951, A061704, A087797, A249770.

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

a[n_] := Times @@ (Round[(# + 3)^2/12]& /@ FactorInteger[n][[All, 2]]);
Array[a, 102] (* Jean-François Alcover, Jan 02 2019 *)

a(n) = vecprod(apply(x -> round((x+3)^2/12), factor(n)[, 2])); \\ Amiram Eldar, Jun 12 2025

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

A249771 Irregular triangle read by rows: T(n,k) is the number of Abelian groups of order A025487(n) with k invariant factors (2 <= n, 1 <= k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 3, 3, 2, 1, 1, 1, 3, 2, 1, 2, 2, 1, 1, 1, 1, 1, 1, 3, 4, 3, 2, 1, 1, 1, 5, 2, 2, 1, 3, 1, 3, 3, 2, 1, 1, 1, 1, 3, 5, 1, 2

Offset: 2

Álvar Ibeas, Nov 06 2014

The length of n-th row is A051282(n).

Signatures differing only by a (trailing) list of ones give identical rows.

			First rows:
1;
1,1;
1;
1,1,1;
1,1;
1,2,1,1;
1,1,1;
1;
1,2,2,1,1;
1,3;
...

Álvar Ibeas, Rows n=2..1075, flattened

Refinement of A050360. Last row elements: A249773. Cf. A249770, A052304.

T(n,1) = 1. If k > 1 and the prime signature is (e_1,...,e_s), T(n,k) = Sum(Product(A008284(e_i,k), i in I) * Product(A026820(e_i,k-1), i not in I)), where the sum is taken over nonempty subsets I of {1,...,s}.
T(n,k) = A249770(A025487(n),k).
T(n,1) + T(n,2) = A052304(n).

A264809 Irregular array read by rows: row n contains the invariant factors of every (up to isomorphism) abelian group of order n for n>=2.

Original entry on oeis.org

2, 3, 4, 2, 2, 5, 6, 7, 8, 4, 2, 2, 2, 2, 9, 3, 3, 10, 11, 12, 6, 2, 13, 14, 15, 16, 8, 2, 4, 4, 4, 2, 2, 2, 2, 2, 2, 17, 18, 6, 3, 19, 20, 10, 2, 21, 22, 23, 24, 12, 2, 6, 2, 2, 25, 5, 5, 26, 27, 9, 3, 3, 3, 3, 28, 14, 2, 29, 30

Offset: 2

Geoffrey Critzer, Nov 25 2015

Every finite abelian group can be uniquely expressed as the direct product: C_n1 X C_n2 X ... X C_ns for some integers n1,n2,...,ns where each integer is greater than 1 and each successive integer divides its predecessor. The integers n1,n2,...,ns are called the invariant factors. The order of the group is the product of its invariant factors.

			{2},
{3},
{4}, {2, 2},
{5},
{6},
{7},
{8}, {4, 2}, {2, 2, 2},
{9}, {3, 3},
{10},
{11},
{12}, {6, 2},
{13},
{14},
{15},
{16}, {8, 2}, {4, 4}, {4, 2, 2}, {2, 2, 2, 2}
{17},
{18}, {6, 3},
{19},
{20}, {10, 2},
{21},
{22},
{23},
{24}, {12, 2}, {6, 2, 2},
{25}, {5, 5},
{26},
{27}, {9, 3}, {3, 3, 3},
{28}, {14, 2},
{29},
{30},
The row corresponding to n = 12 is 12,6,2 because the invariant factor decompositions of the 2, A000688(12), abelian groups of order 12 are: C_12 and C_6 X C_2

D. S. Dummit and R. M. Foote, Abstract Algebra, Wiley, 2003, 3rd Edition, page 158.

Alois P. Heinz, Rows n = 2..5000, flattened

Cf. A249770.

f[{x_, y_}] := x^IntegerPartitions[y];
g[n_] := FactorInteger[n][[1, 1]];
h[list_] :=Apply[Times,Map[PadRight[#, Max[Map[Length, SplitBy[list, g]]], 1] &,SplitBy[list, g]]];
Table[Map[h, Join @@@ Tuples[Map[f, FactorInteger[n]]]], {n, 2,
   30}] // Grid

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

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A249771 Irregular triangle read by rows: T(n,k) is the number of Abelian groups of order A025487(n) with k invariant factors (2 <= n, 1 <= k).

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Formula

A264809 Irregular array read by rows: row n contains the invariant factors of every (up to isomorphism) abelian group of order n for n>=2.

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Mathematica