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

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

1,11

Comments

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).

Links

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

Formula

(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}.

Example

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.

Crossrefs

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

Sequence in context: A322127 A282496 A253238 * A030369 A298667 A226166

Adjacent sequences: A249770 A249771 A249772 * A249774 A249775 A249776

Keyword

nonn

Author

Álvar Ibeas, Nov 07 2014

Status

approved