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

Author

Álvar Ibeas, Nov 07 2014

Keywords

• nonn

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).

Examples

			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.
		

Links

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

Crossrefs

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

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}.