A265120 Irregular array read by rows: Row n gives the number of elements in the multiplicative group mod n, (Z/nZ, *), that have order d for each divisor d of the exponent of the group.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 3, 1, 1, 2, 2, 1, 1, 2, 1, 1, 4, 4, 1, 3, 1, 1, 2, 2, 2, 4, 1, 1, 2, 2, 1, 3, 4, 1, 3, 4, 1, 1, 2, 4, 8, 1, 1, 2, 2, 1, 1, 2, 2, 6, 6, 1, 3, 4, 1, 3, 2, 6, 1, 1, 4, 4, 1, 1, 10, 10, 1, 7, 1, 1, 2, 4, 4, 8

Offset

1,9

Comments

The exponent of the multiplicative group mod n is Carmichael lambda(n) given in A002322.

The row lengths are tau(lambda(n)) = A000005(A002322(n)) = A066800(n).

The invariant factor decomposition of (Z/nZ,*) is given in A258446.

The row sums are phi(n) = A000010(n).

It appears that column 2 is A155828.

Links

Jianing Song, Table of n, a(n) for n = 1..9396 (rows 1..1000)

Example

{1}
{1}
{1, 1}
{1, 1}
{1, 1, 2}
{1, 1}
{1, 1, 2, 2}
{1, 3}
{1, 1, 2, 2}
{1, 1, 2}
{1, 1, 4, 4}
{1, 3}
{1, 1, 2, 2, 2, 4}
{1, 1, 2, 2}
{1, 3, 4}
{1, 3, 4}
{1, 1, 2, 4, 8}
{1, 1, 2, 2}
{1, 1, 2, 2, 6, 6}
{1, 3, 4}
{1, 3, 2, 6}
{1, 1, 4, 4}
{1, 1, 10, 10}
{1, 7},
{1, 1, 2, 4, 4, 8}
The row for n=21 reads: 1,3,2,6 because the multiplicative group mod 21,  (Z/21*Z,*) is isomorphic to C_6 X C_2. The exponent of this group is 6. This group contains one element of order 1, three elements of order 2, two elements of order 3, and six elements of order 6.

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 &];
t[list_] :=
Table[Count[Map[PermutationOrder, GroupElements[AbelianGroup[list]]],
    d], {d, Divisors[First[list]]}];
Map[t, Table[
   If[! IntegerQ[n/8],
    DeleteCases[rec[Flatten[Map[f, FactorInteger[n]]]], 0|1],
    DeleteCases[
     rec[Join[{2, 2^(FactorInteger[n][[1, 2]] - 2)},
       Flatten[Map[f, Drop[FactorInteger[n], 1]]]]], 1]], {n, 1, 25}] /. {} -> {1}] (* Edited by Jianing Song, May 29 2026 to include n = 1 *)

Prog

(PARI) power_solution(invariant_factors, k) = vecprod(apply(x->gcd(x, k), invariant_factors)); \\ number of solutions to x^k = 1 in C_{k_1} X ... C_{k_r}
count_order(invariant_factors, k) = sumdiv(k, d, moebius(k/d)*power_solution(invariant_factors, d)); \\ number of elements of order k
row(n) = my(invariant_factors=znstar(n)[2], k=lcm(invariant_factors)); apply(x->count_order(invariant_factors, x), divisors(k)); \\ Jianing Song, May 29 2026

Crossrefs

Cf. A000005, A000010, A002322, A066800, A155828, A258446.

Cf. A111725 (rightmost elements of rows).

Sequence in context: A296773 A108244 A277824 * A329621 A124961 A373269

Adjacent sequences: A265117 A265118 A265119 * A265121 A265122 A265123

Keyword

nonn,tabf,changed

Author

Geoffrey Critzer, Dec 01 2015

Status

approved