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%I #12 Sep 19 2023 10:24:01
%S 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,
%T 7,4,3,3,4,16,7,10,4,7,4,16,3,3,4,13,3,4
%N Number of independent cycles of the multiplicative group of integers modulo A033949(n).
%C 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.
%C See Table 7, column 4 of the W. Lang link for these numbers.
%C See also the Table in the W. Lang link given in A282624 for these independent cycles.
%H Wolfdieter Lang, <a href="http://arxiv.org/abs/1210.1018">The field Q(2cos(pi/n)), its Galois group and length ratios in the regular n-gon</a>, arXiv:1210.1018 [math.GR], 2012-2017.
%e 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.
%e 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.
%Y Cf. A033949, A282624.
%K nonn,more
%O 1,1
%A _Wolfdieter Lang_, Mar 03 2017
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