%I
%S 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,
%T 45,20,77,60,55,18,66,28,49,101,84,75,30,90,44,77,135,112,110,42,27,
%U 126,60,105,50,176,154,150,66,16,121,45,168,88,154,70,231,202
%N a(n) = number of Abelian groups of order A181800(n) (nth powerful number that is the first integer of its prime signature).
%C 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.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AbelianGroup.html">Abelian Group</a>
%F a(n) = A000688(A181800(n)).
%e 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.
%Y Cf. A000688, A046054, A046055, A046056, A050360, A181800, A212172.
%K nonn
%O 1,2
%A _Matthew Vandermast_, Jun 09 2012
