A282624 -id:A282624 - cp's OEIS frontend

A281854 Irregular triangle read by rows. Row n gives the orders of the cyclic groups appearing as factors in the direct product decomposition of the abelian non-cyclic multiplicative groups of integers modulo A033949(n).

2, 2, 2, 2, 4, 2, 4, 2, 4, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 4, 2, 8, 2, 5, 2, 2, 4, 3, 2, 3, 2, 2, 4, 3, 2, 4, 2, 2, 3, 2, 2, 5, 2, 2, 4, 3, 2, 4, 2, 2, 16, 2, 4, 3, 2, 5, 4, 2, 3, 2, 2, 2, 9, 2, 2, 4, 2, 2

Offset: 1

Wolfdieter Lang, Feb 02 2017

The length of row n is given in A281855.
The multiplicative group of integers modulo n is written as (Z/(n Z))^x (in ring notation, group of units) isomorphic to Gal(Q(zeta(n))/Q) with zeta(n) = exp(2*Pi*I/n). The present table gives in row n the factors of the direct product decomposition of the non-cyclic group of integers modulo A033949(n) (in nonincreasing order). The cyclic group of order n is C_n. Note that only C-factors of prime power orders are used; for example C_6 has the decomposition C_3 x C_2, etc. C_n is decomposed whenever n has relatively prime factors like in C_30 = C_15 x C_2 = C_5 x C_3 x C_2. In the Wikipedia table partial decompositions appear.
The row products phi(A033949(n)) are given as 4*A281856(n), n >= 1, with phi(n) = A000010(n).
See also the W. Lang links for these groups.

			The triangle T(n, k) begins (N = A033949(n)):
n,   N, phi(N)\ k  1  2  3  4 ...
1,   8,   4:       2  2
2,  12,   4:       2  2
3,  15,   8:       4  2
4,  16,   8:       4  2
5,  20,   8:       4  2
6,  21,  12:       3  2  2
7,  24,   8:       2  2  2
8,  28,  12:       3  2  2
9,  30,   8:       4  2
10, 32,  16:       8  2
11, 33,  20:       5  2  2
12, 35,  24:       4  3  2
13, 36,  12:       3  2  2
14, 39,  24:       4  3  2
15, 40,  16:       4  2  2
16, 42,  12:       3  2  2
17, 44,  20:       5  2  2
18, 45,  24:       4  3  2
19, 48,  16:       4  2  2
20, 51,  32:      16  2
21, 52,  24:       4  3  2
22, 55,  40:       5  4  2
23, 56,  24:       3  2  2  2
24, 57,  36:       9  2  2
25, 60,  16:       4  2  2
...
n = 6, A033949(6) = N = 21, phi(21) = 12, group (Z/21 n)^x decomposition C_3 x C_2 x C_2 (in the Wikipedia Table C_2 x C_6). The smallest positive reduced system modulo 21 has the primes {2, 5, 11, 13, 17, 19} with cycle lengths {6, 6, 6, 2, 6, 6}, respectively. As generators of the group one can take <2, 13>.
  (In the Wikipedia Table <2, 20> is used).
----------------------------------------------
From _Wolfdieter Lang_, Feb 04 2017: (Start)
n = 32, A033949(32) = N = 70, phi(70) = 24.
Cycle types (multiplicity as subscript): 12_7, 6_4, 4_2, 3_1, 2_2 (a total of 16 cycles). Cycle structure: 12_2, 6_2 (all other cycles are sub-cycles).
The first 12-cycle obtained from the powers of, say 3, contains also the 12-cycles from 17 and 47. It also contains the 4-cycle from 13, the 3-cycle from 11 and the 2-cycle from 29.
The second 12-cycle from the powers of, say, 23 contains also the 12-cycles from 37, 53 and 67, as well as the 4-cycle from 43.
The first 6-cycle from the powers of, say, 19 contains also the 6-cycle of 59 as well as the 2-cycle from 41.
The second 6-cycle from the powers of, say, 31 contains also the 6-cycle from 61.
The group is C_6 x C_4 = (C_2 x C_3) x C_4 = C_4 X C_3 x C_2 (see the W. Lang link, Table 7)
The cycle graph of C_4 X C_3 x C_2 is the 7th entry of Figure 4 of this link.
(End)

Wolfdieter Lang, The field Q(2cos(pi/n)), its Galois group and length ratios in the regular n-gon, Table 7 (in row n = 80 it should read Z_4^2 x Z_2), arXiv:1210.1018 [math.GR], 2012.
Wolfdieter Lang, Table for the multiplicative non-cyclic groups of integers modulo A033949.
Wikipedia, Multiplicative group of integers modulo n . Compare with the Table at the end.

Cf. A033949, A192005, A281855, A282624.

A281855 Number of cyclic group factors in the total decomposition of the abelian non-cyclic group (Z/A033949(n) Z)^x. Row lengths of A281854.

Original entry on oeis.org

2, 2, 2, 2, 2, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 4, 3, 3, 4, 2, 3, 3, 2, 3, 3, 4, 3, 3, 4, 3, 3, 4, 2, 3, 4, 3, 4, 3, 4, 3, 3, 4, 3, 2, 4, 4, 3, 3, 3, 4, 3, 3, 3, 4, 3, 4, 3, 4, 4, 4, 2, 4, 3

Offset: 1

Wolfdieter Lang, Mar 03 2017

See A281854.

Compare this with A046072(A033949(n)). In A046072 the decompositions used are not total, e.g., n = 6 with A033949(6) = 1 uses C_6, but C_6 = C_2 x C_3. Or, 21 = A033949(6), A046072(21) = 2 not 3 = a(6).

Cf. A033949, A046072, A281854, A282623, A282624, A282625.

A282625 Number of cyclic groups in the total direct product factorization of the multiplicative group of integers modulo n, for n >= 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 2, 3, 2, 3, 1, 3, 3, 2, 2, 3, 3, 2, 3, 3, 3, 3, 2, 2, 3, 3, 2, 2, 3, 2, 2, 3, 4, 3, 2, 2, 3, 3, 3, 4, 2, 3, 3, 3, 2, 3, 3, 3, 4, 2, 2, 3, 3, 4, 3, 3, 3, 2, 2, 2, 4, 2, 3, 3, 4, 2, 3, 4, 3, 4, 2, 3, 3, 2, 3, 4, 3

Offset: 1

Wolfdieter Lang, Mar 02 2017

nonn

The multiplicative group of integers modulo n, (Z/n*Z)^x, also the cyclotomic group, the Galois group Gal(Q(zeta(n))/Q) with zeta(n) = exp(2*Pi*I/n), is cyclic for n from A033948 and non-cyclic for n from A033949. Each of these groups is the direct product of cyclic factors (one factor is included).
In the total factorization for n >= 3 only cyclic factors whose orders are prime powers appear, and because the direct product is associative, and for these abelian groups also commutative, one can order the factors with nonincreasing orders.
For n=1 and n=2 the group is C_1 = {1} (for n=1 one has 1 == 0 (mod 1)).
Cyclic groups may also have a factorization into more than one factor. E.g., C_6 = C_3 x C_2.
The number of factors in this total factorization is for a cyclic group C_m, for m >= 2, given by A001221(m). For m=1 this number is 1 (not A001221(1)).
For non-cyclic groups the number of factors in this total factorization is given by A281855(m) if n = A033949(m), m >= 1.
For the non-cyclic group case see also the W. Lang links under A281854.
Compare this sequence with A046072 where another factorization of these groups is used, the one with the least cyclic factors. E.g., A046072(7) = 1 for the group C_6, and a(7) = 2 here (see the example above).

			n = 35, a non-cyclic case because A033949(12) = 35. The group can be written as <19_6, 13_4 > where the orders modulo 35 of the generators are given as subscript. Therefore the group is C_6 x C4 = C_4 x C_3 x C_2 and a(35) = 3, whereas A046072(35) = 2.

Cf. A001221, A033948, A033949, A046072, A281854, A281855, A282624.

A282623 Number of independent cycles of the multiplicative group of integers modulo A033949(n).

Original entry on oeis.org

3, 3, 4, 4, 4, 3, 7, 3, 4, 5, 3, 4, 3, 4, 10, 3, 3, 4, 10, 6, 4, 4, 7, 3, 10, 12, 6, 6, 3, 6, 3, 4, 7, 4, 3, 3, 4, 16, 7, 10, 4, 7, 4, 16, 3, 3, 4, 13, 3, 4

Offset: 1

Wolfdieter Lang, Mar 03 2017

A cycle starting with number a of the restricted residue system modulo m (namely the one with the smallest positive numbers RRS(m)) is independent of a cycle starting with number b != a if the set of numbers of the a-cycle is not a (not necessarily proper) subset of the numbers of the b-cycle.
See Table 7, column 4 of the W. Lang link for these numbers.
See also the Table in the W. Lang link given in A282624 for these independent cycles.

			a(1) = 3 because A033949(1) = 8 with RRS(8) = {1, 3, 5, 7} and the three 2-cycles [3,1],[5,1] and [7,1], which are independent.
a(4) = 4 because A033949(4) = 16 with RRS(16) = {1, 3, 5, 7, 9, 11, 13, 15} and only, e.g., the cycles from 3, 5, 7 and 15 are independent. The cycles [1], [9, 1], [11, 9, 3, 1] and [13, 9, 5, 1] are not independent. One could replace 5 with 13 but we always take the smallest numbers.

Wolfdieter Lang, The field Q(2cos(pi/n)), its Galois group and length ratios in the regular n-gon, arXiv:1210.1018 [math.GR], 2012-2017.

Cf. A033949, A282624.

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A281854 Irregular triangle read by rows. Row n gives the orders of the cyclic groups appearing as factors in the direct product decomposition of the abelian non-cyclic multiplicative groups of integers modulo A033949(n).

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A281855 Number of cyclic group factors in the total decomposition of the abelian non-cyclic group (Z/A033949(n) Z)^x. Row lengths of A281854.

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A282625 Number of cyclic groups in the total direct product factorization of the multiplicative group of integers modulo n, for n >= 1.

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A282623 Number of independent cycles of the multiplicative group of integers modulo A033949(n).

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