

A216320


Irregular triangle: row n lists the Modd n order of the odd members of the reduced smallest nonnegative residue class modulo n.


5



1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 3, 1, 4, 4, 2, 1, 3, 3, 1, 4, 4, 2, 1, 5, 5, 5, 5, 1, 2, 2, 2, 1, 3, 2, 6, 3, 6, 1, 3, 6, 3, 6, 2, 1, 4, 2, 4, 1, 8, 8, 4, 4, 8, 8, 2, 1, 8, 8, 8, 4, 8, 2, 4, 1, 6, 6, 3, 3, 2, 1, 9, 9, 3, 9, 3, 9, 9, 9, 1, 4, 4, 2, 2, 4, 4, 2, 1, 3, 6, 2, 3, 6
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OFFSET

1,5


COMMENTS

The length of row n is delta(n):=A055034(n).
For the multiplicative group Modd n see a comment on A203571, and also on A216319.
A216319(n,k)^a(n,k) == +1 (Modd n), n >= 1.
If the Modd n order of an (odd) element from row n of A216319 is delta(n) (the row length) then this element is a primitive root Modd n. There is no primitive root Modd n if no such element of order delta(n) exists. For example, n = 12, 20, ... (see A206552 for more of these n values). There are phi(delta(n)) = A216321(n) such primitive roots Modd n if there exists one, where phi=A000010 (Euler's totient). The multiplicative group Modd n is cyclic if and only if there exists a primitive root Modd n. The multiplicative group Modd n is isomorphic to the Galois group G(Q(rho(n)/Q) with the algebraic number rho(n) := 2*cos(Pi/n), n>=1.


LINKS



FORMULA

a(n,k) = order of A216319(n,k) Modd n, n>=1, k=1, 2, ..., A055034(n). This means: A216319(n,k)^a(n,k) == +1 (Modd n), n>=1, and a(n,k) is the smallest positive integer exponent satisfying this congruence. For Modd n see a comment on A203571.


EXAMPLE

The table a(n,k) begins:
n\k 1 2 3 4 5 6 7 8 9 ...
1 1
2 1
3 1
4 1 2
5 1 2
6 1 2
7 1 3 3
8 1 4 4 2
9 1 3 3
10 1 4 4 2
11 1 5 5 5 5
12 1 2 2 2
13 1 3 2 6 3 6
14 1 3 6 3 6 2
15 1 4 2 4
16 1 8 8 4 4 8 8 2
17 1 8 8 8 4 8 2 4
18 1 6 6 3 3 2
19 1 9 9 3 9 3 9 9 9
20 1 4 4 2 2 4 4 2
...
a(7,2) = 3 because A216319(7,2) = 3 and 3^1 == 3 (Modd 7);
3^2 = 9 == 5 (Modd 7) because floor(9/7)= 1 which is odd, therefore 9 (Modd 7) = 9 (mod 7) = 5; 3^3 == 5*3 (Modd n)
= +1 because floor(15/7)=2 which is even, therefore 15 (Modd 7) = 15 (modd 7) = +1.
Row n=12 is the first row without an order = delta(n) (row length), in this case 4. Therefore there is no primitive root Modd 12, and the multiplicative group Modd 12 is noncyclic.
Its cycle structure is [[5,1],[7,1],[11,1]] which is the group Z_2 x Z_2 (the Klein 4group).


CROSSREFS



KEYWORD

nonn,easy,tabf


AUTHOR



STATUS

approved



