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%I #15 Nov 27 2015 06:30:31
%S 2,3,4,2,2,5,6,7,8,4,2,2,2,2,9,3,3,10,11,12,6,2,13,14,15,16,8,2,4,4,4,
%T 2,2,2,2,2,2,17,18,6,3,19,20,10,2,21,22,23,24,12,2,6,2,2,25,5,5,26,27,
%U 9,3,3,3,3,28,14,2,29,30
%N Irregular array read by rows: row n contains the invariant factors of every (up to isomorphism) abelian group of order n for n>=2.
%C Every finite abelian group can be uniquely expressed as the direct product: C_n1 X C_n2 X ... X C_ns for some integers n1,n2,...,ns where each integer is greater than 1 and each successive integer divides its predecessor. The integers n1,n2,...,ns are called the invariant factors. The order of the group is the product of its invariant factors.
%D D. S. Dummit and R. M. Foote, Abstract Algebra, Wiley, 2003, 3rd Edition, page 158.
%H Alois P. Heinz, <a href="/A264809/b264809.txt">Rows n = 2..5000, flattened</a>
%e {2},
%e {3},
%e {4}, {2, 2},
%e {5},
%e {6},
%e {7},
%e {8}, {4, 2}, {2, 2, 2},
%e {9}, {3, 3},
%e {10},
%e {11},
%e {12}, {6, 2},
%e {13},
%e {14},
%e {15},
%e {16}, {8, 2}, {4, 4}, {4, 2, 2}, {2, 2, 2, 2}
%e {17},
%e {18}, {6, 3},
%e {19},
%e {20}, {10, 2},
%e {21},
%e {22},
%e {23},
%e {24}, {12, 2}, {6, 2, 2},
%e {25}, {5, 5},
%e {26},
%e {27}, {9, 3}, {3, 3, 3},
%e {28}, {14, 2},
%e {29},
%e {30},
%e The row corresponding to n = 12 is 12,6,2 because the invariant factor decompositions of the 2, A000688(12), abelian groups of order 12 are: C_12 and C_6 X C_2
%t f[{x_, y_}] := x^IntegerPartitions[y];
%t g[n_] := FactorInteger[n][[1, 1]];
%t h[list_] :=Apply[Times,Map[PadRight[#, Max[Map[Length, SplitBy[list, g]]], 1] &,SplitBy[list, g]]];
%t Table[Map[h, Join @@@ Tuples[Map[f, FactorInteger[n]]]], {n, 2,
%t 30}] // Grid
%Y Cf. A249770.
%K nonn,tabf
%O 2,1
%A _Geoffrey Critzer_, Nov 25 2015
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