A281856 -id:A281856 - cp's OEIS frontend

A282624 Irregular triangle read by rows: row n gives a certain choice of generators of the multiplicative group of integers modulo A033949(n).

3, 5, 5, 7, 2, 11, 3, 7, 3, 11, 2, 13, 5, 7, 13, 3, 13, 7, 11, 3, 31, 2, 23, 19, 13, 5, 19, 17, 5, 3, 11, 29, 5, 13, 3, 43, 11, 17, 5, 7, 17, 5, 35, 3, 5, 19, 23, 3, 13, 29, 2, 37, 7, 11, 19, 2, 5, 3, 31, 2, 31, 5, 43, 3, 67, 2, 68, 19, 13, 5, 17, 19, 11, 7

Offset: 1

Wolfdieter Lang, Mar 03 2017

The length of row n is given by A046072(A033949(n)), n >= 1.
The generators are chosen minimally in the sense that the product of their orders (cycle lengths) is phi(N(n)) = A000010(N(n)) with N(n) = A033949(n). In addition, the generators are sorted with nonincreasing orders, and the smallest numbers with these orders are listed.
Note that the first instance where a composite generator is needed is N = 51 = A033949(20) with a generator 35. The next such number is N = 69 = A033949(31) with a generator 68. Such numbers N will be called exceptional.
For a table with n = 1..69, N = 8, 12, ..., 130, see the W. Lang link. Compare this with the Wikipedia table (where some generator errors will be corrected). There non-minimal generators are also used, i.e., the product of the orders of the generators is larger than phi(N). The Wikipedia table often uses composite generators when primes would do the job. E.g., N = 16 with generators 2, 14 instead of 2, 11; or N = 16 with 3, 15 instead of 3, 7, etc.

			The irregular triangle T(n, k) begins (here N = A033949(n), and the respective primitive cycle lengths and phi(N) are also given)
n,   N \k 1   2   3 ... cycle lengths, phi(N)
1,   8:   3   5           2  2          4
2,  12:   5   7           2  2          4
3,  15:   2  11           4  2          8
4,  16:   3   7           4  2          8
5,  20:   3  11           4  2          8
6,  21:   2  13           6  2         12
7,  24:   5   7  13       2  2  2       8
8,  28:   3  13           6  2         12
9,  30:   7  11           4  2          8
10, 32:   3  31           8  2         16
11, 33:   2  23          10  2         20
12, 35:  19  13           6  4         24
13, 36:   5  19           6  2         12
14, 39:  17   5           6  4         24
15, 40:   3  11  29       4  2  2      16
16: 42:   5  13           6  2         12
17, 44:   3  43          10  2         20
18, 45:  11  17           6  4         24
19, 48:   5   7  17       4  2  2      16
20, 51:   5  35          16  2         32
... See the link for more.

Wolfdieter Lang, Table for the multiplicative non-cyclic groups of integers modulo A033949.
Eldar Sultanow, Christian Koch, and Sean Cox, Collatz Sequences in the Light of Graph Theory, Universität Potsdam (Germany, 2020).

Cf. A033949, A046072, A279399, A281855, A281856, A282623, A282625.

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).

Original entry on oeis.org

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.

cp's OEIS Frontend

A282624 Irregular triangle read by rows: row n gives a certain choice of generators of the multiplicative group of integers modulo A033949(n).

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