%I #38 Nov 13 2017 02:58:57
%S 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,
%T 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
%N 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).
%C The length of row n is given in A281855.
%C 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.
%C The row products phi(A033949(n)) are given as 4*A281856(n), n >= 1, with phi(n) = A000010(n).
%C See also the W. Lang links for these groups.
%H Wolfdieter Lang, <a href="https://arxiv.org/abs/1210.1018">The field Q(2cos(pi/n)), its Galois group and length ratios in the regular n-gon</a>, Table 7 (in row n = 80 it should read Z_4^2 x Z_2), arXiv:1210.1018 [math.GR], 2012.
%H Wolfdieter Lang, <a href="/A282624/a282624.pdf">Table for the multiplicative non-cyclic groups of integers modulo A033949</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Multiplicative_group_of_integers_modulo_n ">Multiplicative group of integers modulo n </a>. Compare with the Table at the end.
%e The triangle T(n, k) begins (N = A033949(n)):
%e n, N, phi(N)\ k 1 2 3 4 ...
%e 1, 8, 4: 2 2
%e 2, 12, 4: 2 2
%e 3, 15, 8: 4 2
%e 4, 16, 8: 4 2
%e 5, 20, 8: 4 2
%e 6, 21, 12: 3 2 2
%e 7, 24, 8: 2 2 2
%e 8, 28, 12: 3 2 2
%e 9, 30, 8: 4 2
%e 10, 32, 16: 8 2
%e 11, 33, 20: 5 2 2
%e 12, 35, 24: 4 3 2
%e 13, 36, 12: 3 2 2
%e 14, 39, 24: 4 3 2
%e 15, 40, 16: 4 2 2
%e 16, 42, 12: 3 2 2
%e 17, 44, 20: 5 2 2
%e 18, 45, 24: 4 3 2
%e 19, 48, 16: 4 2 2
%e 20, 51, 32: 16 2
%e 21, 52, 24: 4 3 2
%e 22, 55, 40: 5 4 2
%e 23, 56, 24: 3 2 2 2
%e 24, 57, 36: 9 2 2
%e 25, 60, 16: 4 2 2
%e ...
%e 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>.
%e (In the Wikipedia Table <2, 20> is used).
%e ----------------------------------------------
%e From _Wolfdieter Lang_, Feb 04 2017: (Start)
%e n = 32, A033949(32) = N = 70, phi(70) = 24.
%e 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).
%e 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.
%e 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.
%e 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.
%e The second 6-cycle from the powers of, say, 31 contains also the 6-cycle from 61.
%e 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)
%e The cycle graph of C_4 X C_3 x C_2 is the 7th entry of Figure 4 of this link.
%e (End)
%Y Cf. A033949, A192005, A281855, A282624.
%K nonn,tabf
%O 1,1
%A _Wolfdieter Lang_, Feb 02 2017