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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). 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..68.

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.

EXAMPLE

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)

CROSSREFS

Cf. A033949, A192005, A281855, A282624.

Sequence in context: A046927 A084718 A154851 * A335385 A037445 A318307

Adjacent sequences:  A281851 A281852 A281853 * A281855 A281856 A281857

KEYWORD

nonn,tabf

AUTHOR

Wolfdieter Lang, Feb 02 2017

STATUS

approved

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Last modified June 17 14:24 EDT 2021. Contains 345083 sequences. (Running on oeis4.)