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%I #16 Dec 04 2015 13:04:06
%S 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,
%T 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,
%U 1,4,4,1,1,10,10,1,7,1,1,2,4,4,8
%N 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.
%C The exponent of the multiplicative group mod n is Carmichael lambda(n) given in A002322.
%C The row lengths are tau(lambda(n)) = A000005(A002322(n)) = A066800(n).
%C The invariant factor decomposition of (Z/nZ,*) is given in A258446.
%C The row sums are phi(n) = A000010(n).
%C It appears that column 2 is A155828.
%e {1}
%e {1, 1}
%e {1, 1}
%e {1, 1, 2}
%e {1, 1}
%e {1, 1, 2, 2}
%e {1, 3}
%e {1, 1, 2, 2}
%e {1, 1, 2}
%e {1, 1, 4, 4}
%e {1, 3}
%e {1, 1, 2, 2, 2, 4}
%e {1, 1, 2, 2}
%e {1, 3, 4}
%e {1, 3, 4}
%e {1, 1, 2, 4, 8}
%e {1, 1, 2, 2}
%e {1, 1, 2, 2, 6, 6}
%e {1, 3, 4}
%e {1, 3, 2, 6}
%e {1, 1, 4, 4}
%e {1, 1, 10, 10}
%e {1, 7},
%e {1, 1, 2, 4, 4, 8}
%e 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.
%t f[{p_, e_}] := {FactorInteger[p - 1][[All, 1]]^
%t FactorInteger[p - 1][[All, 2]],
%t FactorInteger[p^(e - 1)][[All, 1]]^
%t FactorInteger[p^(e - 1)][[All, 2]]};
%t fun[lst_] :=
%t Module[{int, num, res},
%t int = Sort /@ GatherBy[Join @@ (FactorInteger /@ lst), First];
%t num = Times @@ Power @@@ (Last@# & /@ int);
%t res = Flatten[Map[Power @@ # &, Most /@ int, {2}]];
%t {num, res}]
%t rec[lt_] :=
%t First@NestWhile[{Append[#[[1]], fun[#[[2]]][[1]]],
%t fun[#[[2]]][[2]]} &, {{}, lt}, Length[#[[2]]] > 0 &];
%t t[list_] :=
%t Table[Count[Map[PermutationOrder, GroupElements[AbelianGroup[list]]],
%t d], {d, Divisors[First[list]]}];
%t Map[t, Table[
%t If[! IntegerQ[n/8],
%t DeleteCases[rec[Flatten[Map[f, FactorInteger[n]]]], 1],
%t DeleteCases[
%t rec[Join[{2, 2^(FactorInteger[n][[1, 2]] - 2)},
%t Flatten[Map[f, Drop[FactorInteger[n], 1]]]]], 1]], {n, 2,
%t 25}] /. {} -> {1}]
%Y Cf. A000005, A000010, A002322, A066800, A155828, A258446.
%K nonn,tabf
%O 2,8
%A _Geoffrey Critzer_, Dec 01 2015
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