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

%I A282624 #14 Mar 05 2021 21:45:15
%S A282624 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,
%T A282624 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,
%U A282624 2,5,3,31,2,31,5,43,3,67,2,68,19,13,5,17,19,11,7
%N A282624 Irregular triangle read by rows: row n gives a certain choice of generators of the multiplicative group of integers modulo A033949(n).
%C A282624 The length of row n is given by A046072(A033949(n)), n >= 1.
%C A282624 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.
%C A282624 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.
%C A282624 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.
%H A282624 Wolfdieter Lang, <a href="/A282624/a282624.pdf">Table for the multiplicative non-cyclic groups of integers modulo A033949</a>.
%H A282624 Eldar Sultanow, Christian Koch, and Sean Cox, <a href="https://doi.org/10.25932/publishup-48214">Collatz Sequences in the Light of Graph Theory</a>, Universität Potsdam (Germany, 2020).
%e A282624 The irregular triangle T(n, k) begins (here N = A033949(n), and the respective primitive cycle lengths and phi(N) are also given)
%e A282624 n,   N \k 1   2   3 ... cycle lengths, phi(N)
%e A282624 1,   8:   3   5           2  2          4
%e A282624 2,  12:   5   7           2  2          4
%e A282624 3,  15:   2  11           4  2          8
%e A282624 4,  16:   3   7           4  2          8
%e A282624 5,  20:   3  11           4  2          8
%e A282624 6,  21:   2  13           6  2         12
%e A282624 7,  24:   5   7  13       2  2  2       8
%e A282624 8,  28:   3  13           6  2         12
%e A282624 9,  30:   7  11           4  2          8
%e A282624 10, 32:   3  31           8  2         16
%e A282624 11, 33:   2  23          10  2         20
%e A282624 12, 35:  19  13           6  4         24
%e A282624 13, 36:   5  19           6  2         12
%e A282624 14, 39:  17   5           6  4         24
%e A282624 15, 40:   3  11  29       4  2  2      16
%e A282624 16: 42:   5  13           6  2         12
%e A282624 17, 44:   3  43          10  2         20
%e A282624 18, 45:  11  17           6  4         24
%e A282624 19, 48:   5   7  17       4  2  2      16
%e A282624 20, 51:   5  35          16  2         32
%e A282624 ... See the link for more.
%Y A282624 Cf. A033949, A046072, A279399, A281855, A281856, A282623, A282625.
%K A282624 nonn,tabf
%O A282624 1,1
%A A282624 _Wolfdieter Lang_, Mar 03 2017