A258446 Irregular triangular array read by rows. Row n gives the decomposition of the multiplicative group of integers modulo n as a direct product of cyclic groups C_1 X C_2 X ... X C_k, where |C_i| divides |C_j|, i>j.

1, 2, 2, 4, 2, 6, 2, 2, 6, 4, 10, 2, 2, 12, 6, 4, 2, 4, 2, 16, 6, 18, 4, 2, 6, 2, 10, 22, 2, 2, 2, 20, 12, 18, 6, 2, 28, 4, 2, 30, 8, 2, 10, 2, 16, 12, 2, 6, 2, 36, 18, 12, 2, 4, 2, 2, 40, 6, 2, 42, 10, 2, 12, 2, 22, 46, 4, 2, 2, 42, 20

Offset: 2

Author

Geoffrey Critzer and Mark Dooris, May 30 2015

Keywords

• nonn
• tabf

Comments

Row lengths are A046072.

The products of the terms in each row are A000010.

First column is A002322.

Row 2^k is [2, 2^(k-2)] for k > 2. - Tom Edgar, May 31 2015

Row p^k (and row 2*p^k) is [(p-1)*p^(k-1)] for odd prime p. - Tom Edgar, May 31 2015

The number of distinct groups over numbers less than or equal to 10^k for k=1,2,3,4,5,6 is 5, 50, 447, 4060, 36655, 335714.

Examples

			1;
2;
2;
4;
2;
6;
2, 2;
6;
4;
10;
2, 2;
12;
6;
4, 2;
4, 2;
16;
6;
18;
4, 2;
6, 2;
10;
22;
2, 2, 2;
The row for n=8 reads: 2,2 because the multiplicative group mod 8 is isomorphic to C_2 X C_2.
		

References

• Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 92-93, 1993.

Links

• Eric Weisstein's World of Mathematics, Modulo Multiplication Group
• Wikipedia, Multiplicative Group of Integers Modulo n.

Programs

Mathematica

f[{p_, e_}] := {FactorInteger[p - 1][[All, 1]]^
   FactorInteger[p - 1][[All, 2]],
  FactorInteger[p^(e - 1)][[All, 1]]^
   FactorInteger[p^(e - 1)][[All, 2]]};
fun[lst_] :=
Module[{int, num, res},
  int = Sort /@ GatherBy[Join @@ (FactorInteger /@ lst), First];
  num = Times @@ Power @@@ (Last@# & /@ int);
  res = Flatten[Map[Power @@ # &, Most /@ int, {2}]];
  {num, res}]
rec[lt_] :=
First@NestWhile[{Append[#[[1]], fun[#[[2]]][[1]]],
     fun[#[[2]]][[2]]} &, {{}, lt}, Length[#[[2]]] > 0 &];
Table[If[! IntegerQ[n/8],
    DeleteCases[rec[Flatten[Map[f, FactorInteger[n]]]], 1],
    DeleteCases[
     rec[Join[{2, 2^(FactorInteger[n][[1, 2]] - 2)},
       Flatten[Map[f, Drop[FactorInteger[n], 1]]]]], 1]], {n, 2,
    50}] /. {} -> {1} // Grid