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Number of Abelian groups that attain the maximum number of invariant factors among those whose order is A025487(n).

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`%I #18 Mar 21 2015 05:16:04
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`%S 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,
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`%T 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,
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`%U 10,2,1,7,9,1,3,4,5,1,13,1,3,2,1,2,5,6
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`%N Number of Abelian groups that attain the maximum number of invariant factors among those whose order is A025487(n).
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`%C The number of invariant factors (i.e., the minimum size of generating sets) of these groups is A051282(n).
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`%C 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).
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`%H Álvar Ibeas, <a href="/A249773/b249773.txt">Table of n, a(n) for n = 1..10000</a>
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`%F (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}.
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`%e 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.
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`%Y Last row elements of A249771. Cf. A025487, A051282.
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`%K nonn
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`%O 1,11
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`%A _Álvar Ibeas_, Nov 07 2014
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