A264809 Irregular array read by rows: row n contains the invariant factors of every (up to isomorphism) abelian group of order n for n>=2.

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, 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, 9, 3, 3, 3, 3, 28, 14, 2, 29, 30

Offset

2,1

Comments

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.

References

D. S. Dummit and R. M. Foote, Abstract Algebra, Wiley, 2003, 3rd Edition, page 158.

Links

Alois P. Heinz, Rows n = 2..5000, flattened

Example

{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, 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}, {9, 3}, {3, 3, 3},
{28}, {14, 2},
{29},
{30},
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

Mathematica

f[{x_, y_}] := x^IntegerPartitions[y];
g[n_] := FactorInteger[n][[1, 1]];
h[list_] :=Apply[Times, Map[PadRight[#, Max[Map[Length, SplitBy[list, g]]], 1] &, SplitBy[list, g]]];
Table[Map[h, Join @@@ Tuples[Map[f, FactorInteger[n]]]], {n, 2,
   30}] // Grid

Crossrefs

Cf. A249770.

Sequence in context: A138136 A056472 A162247 * A035578 A227784 A256445

Adjacent sequences: A264806 A264807 A264808 * A264810 A264811 A264812

Keyword

nonn,tabf

Author

Geoffrey Critzer, Nov 25 2015

Status

approved