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

Author

Matthew Vandermast, Jun 09 2012

Keywords

• nonn

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.

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Abelian Group

Crossrefs

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

Formula

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