

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 noncyclic 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
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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 noncyclic group of integers modulo A033949(n) (in nonincreasing order). The cyclic group of order n is C_n. Note that only Cfactors 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 ngon, 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 noncyclic 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 subcycles).
The first 12cycle obtained from the powers of, say 3, contains also the 12cycles from 17 and 47. It also contains the 4cycle from 13, the 3cycle from 11 and the 2cycle from 29.
The second 12cycle from the powers of, say, 23 contains also the 12cycles from 37, 53 and 67, as well as the 4cycle from 43.
The first 6cycle from the powers of, say, 19 contains also the 6cycle of 59 as well as the 2cycle from 41.
The second 6cycle from the powers of, say, 31 contains also the 6cycle 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



