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

Author

Álvar Ibeas, Nov 06 2014

Keywords

• nonn
• tabf

Comments

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.

Examples

			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;
...
		

Links

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

Crossrefs

Refinement of A000688.

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

Programs

Mathematica

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 *)

Formula

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)