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

Author

Wolfdieter Lang, Mar 03 2017

Keywords

• nonn
• tabf

Comments

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.

Examples

			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.
		

Links

• 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).

Crossrefs

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