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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
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

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.

REFERENCES

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

LINKS

Table of n, a(n) for n=2..72.

Eric Weisstein's World of Mathematics, Modulo Multiplication Group

Wikipedia, Multiplicative Group of Integers Modulo n.

EXAMPLE

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.

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

CROSSREFS

Sequence in context: A286280 A179013 A090397 * A054704 A143525 A086087

Adjacent sequences:  A258443 A258444 A258445 * A258447 A258448 A258449

KEYWORD

nonn,tabf

AUTHOR

Geoffrey Critzer and Mark Dooris, May 30 2015

STATUS

approved

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Last modified May 23 16:25 EDT 2017. Contains 286925 sequences.