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%I #32 Aug 27 2023 19:28:20
%S 1,4,8,36,16,72,32,900,216,144,64,1800,0,288,128,44100,0,5400,0,3600,
%T 864,256,0,88200,1296,0,27000,7200,0,512,0,5336100,1728,0,2592,264600,
%U 0,0,0,176400,0,1024,0,2304,3456,0,0,10672200,7776,32400,0,0,0,1323000,5184,2048,0,0,0,4608
%N Smallest order for which there are n nonisomorphic finite Abelian groups, or 0 if no such order exists.
%C There is a k: A000688(k)=n if and only if n is product of partition numbers.
%H Charlie Neder, <a href="/A046056/b046056.txt">Table of n, a(n) for n = 1..1000</a>
%H B. Horvat, G. Jaklic and T. Pisanski, <a href="https://arxiv.org/abs/math/0503183">On the number of Hamiltonian groups</a>, arXiv:math/0503183 [math.CO], 2005.
%H Charlie Neder, <a href="/A046056/a046056.py.txt">Python program for computing this sequence</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AbelianGroup.html">Abelian Group</a>.
%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>
%Y Cf. A000041, A000688, A033637, A046054, A046055, A046057.
%K nonn,nice
%O 1,2
%A _Eric W. Weisstein_
%E More terms from _Christian G. Bower_
%E a(20) and a(28) corrected, and a(52)-a(60) from _Charlie Neder_, Jan 17 2019
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