A212646 a(n) = number of Abelian groups of order A181800(n) (n-th powerful number that is the first integer of its prime signature).

1, 2, 3, 5, 7, 4, 11, 6, 15, 10, 9, 22, 14, 15, 30, 22, 21, 8, 42, 30, 25, 33, 12, 56, 44, 35, 45, 20, 77, 60, 55, 18, 66, 28, 49, 101, 84, 75, 30, 90, 44, 77, 135, 112, 110, 42, 27, 126, 60, 105, 50, 176, 154, 150, 66, 16, 121, 45, 168, 88, 154, 70, 231, 202

Offset

1,2

Comments

The number of Abelian groups of order n, or A000688(n), is a function of the second signature of n (cf. A212172). Since A181800 gives the first integer of each second signature, this sequence gives the value of A000688 for each second signature in order of its first appearance.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Abelian Group

Formula

a(n) = A000688(A181800(n)).

Example

There are 6 Abelian groups of order 72, corresponding to the 6 factorizations of 72 into prime powers: 2^3*3^2, 2^3*3*3, 2^2*2*3^2, 2^2*2*3*3, 2*2*2*3^2, and 2*2*2*3*3. Since 72 = A181800(8), a(8) = 6.

Crossrefs

Cf. A000688, A046054, A046055, A046056, A050360, A181800, A212172.

Sequence in context: A372130 A302024 A273665 * A377901 A369515 A103866

Adjacent sequences: A212643 A212644 A212645 * A212647 A212648 A212649

Keyword

nonn

Author

Matthew Vandermast, Jun 09 2012

Status

approved