A249771 -id:A249771 - cp's OEIS frontend

A249770 Irregular triangle read by rows: T(n,k) is the number of Abelian groups of order n with k invariant factors (2 <= n, 1 <= k).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 2

Álvar Ibeas, Nov 06 2014

The length of n-th row is A051903(n) and its last element is A249773(A101296(n)).

T(n,k) depends only on k and the prime signature of n.

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

Álvar Ibeas, Rows n=2..58654, flattened

Refinement of A000688.

Cf. A008284, A051903, A026820, A101296, A249771, A249773, A046951, A264809.

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]]]; t[list_] := Tally[Map[Length, list]][[All, 2]];
Map[t, Table[Map[h, Join @@@ Tuples[Map[f, FactorInteger[n]]]], {n, 2, 50}]] // Grid (* Geoffrey Critzer, Nov 26 2015 *)

T(n,k) = A249771(A101296(n),k).
T(n,1) = 1. If k > 1 and n = Product(p_i^e_i), 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,...,omega(n)}.
If p is prime and gcd(p,n) = 1, T(pn,k) = T(n,k).
Dirichlet g.f. of column sums: zeta(s)zeta(2s)···zeta(ms) = 1 + Sum_{n >= 2} (Sum_{k=1..m} T(n,k)) / n^s.
T(n,1) + T(n,2) = A046951(n)

A249773 Number of Abelian groups that attain the maximum number of invariant factors among those whose order is A025487(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 5, 1, 1, 2, 2, 1, 1, 3, 1, 1, 2, 2, 1, 1, 3, 7, 1, 1, 5, 2, 3, 9, 2, 1, 1, 3, 4, 1, 1, 3, 2, 1, 2, 5, 2, 1, 1, 3, 4, 1, 1, 3, 2, 1, 2, 5, 10, 2, 1, 7, 9, 1, 3, 4, 5, 1, 13, 1, 3, 2, 1, 2, 5, 6

Offset: 1

Álvar Ibeas, Nov 07 2014

nonn

The number of invariant factors (i.e., the minimum size of generating sets) of these groups is A051282(n).

If the n-th and m-th least (according to the ordering of A025487) prime signatures differ only by a (trailing) list of ones, a(n) = a(m).

			A025487(15) = 72. An Abelian group of order 72 can have 1, 2, or 3 invariant factors. The instances of the last case are C18 x C2 x C2 and C6 x C6 x C2, hence a(15)=2.

Álvar Ibeas, Table of n, a(n) for n = 1..10000

Last row elements of A249771. Cf. A025487, A051282.

(p(e_1)^j - (p(e_1)-1)^j) * Product(p(e_i), i=j+1..s), if the prime signature is (e_i, i=1..s) and e_1 = ... = e_j != e_{j+1}.

A322886 Number of 3-generated Abelian groups of order A025487(n).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 3, 1, 5, 4, 4, 2, 7, 6, 5, 3, 8, 8, 4, 7, 1, 9, 4, 10, 10, 6, 8, 2, 12, 5, 12, 14, 8, 10, 3, 15, 8, 7, 14, 9, 16, 4, 16, 10, 12, 4, 21, 12, 8, 16, 12, 20, 1, 6, 20, 14, 14, 5, 24, 16, 10, 19, 15, 24, 2, 8, 28, 18, 16, 16, 8, 16, 7, 30, 20, 9, 12, 25, 21, 21, 28, 3

Offset: 1

Álvar Ibeas, Dec 29 2018

nonn

Groups generated by fewer than 3 elements are not excluded. The number of Abelian groups with 3 invariant factors is a(n) - A052304(n).

Sum of the first three columns from A249771 (for n > 1).

Jean-François Alcover, Table of n, a(n) for n = 1..300

Cf. A052304, A025487, A249771, A322885.

terms = 300; nmax = 15 terms^2;
A025487 = Module[{lpe = {}, ln = {1}}, Do[pe = FactorInteger[n][[All, 2]] // Sort; If[FreeQ[lpe, pe], AppendTo[lpe, pe]; AppendTo[ln, n]], {n, 2, nmax}]; ln];
a322885[n_] := Times @@ (Round[(#+3)^2/12]& /@ FactorInteger[n][[All, 2]]);
a[n_] := a322885[A025487[[n]]];
Array[a, terms] (* Jean-François Alcover, Jan 02 2019, after Robert G. Wilson v in A025487 *)

a(n) = A322885(A025487(n)).

cp's OEIS Frontend

A249770 Irregular triangle read by rows: T(n,k) is the number of Abelian groups of order n with k invariant factors (2 <= n, 1 <= k).

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A249773 Number of Abelian groups that attain the maximum number of invariant factors among those whose order is A025487(n).

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A322886 Number of 3-generated Abelian groups of order A025487(n).

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