A216319 Irregular triangle: row n lists the odd numbers of the reduced residue system modulo n.

1, 1, 1, 1, 3, 1, 3, 1, 5, 1, 3, 5, 1, 3, 5, 7, 1, 5, 7, 1, 3, 7, 9, 1, 3, 5, 7, 9, 1, 5, 7, 11, 1, 3, 5, 7, 9, 11, 1, 3, 5, 9, 11, 13, 1, 7, 11, 13, 1, 3, 5, 7, 9, 11, 13, 15, 1, 3, 5, 7, 9, 11, 13, 15, 1, 5, 7, 11, 13, 17, 1, 3, 5, 7, 9, 11, 13, 15, 17, 1, 3, 7, 9, 11, 13, 17, 19

Offset: 1

Author

Wolfdieter Lang, Sep 21 2012

Keywords

• nonn
• easy
• tabf

Comments

The length of row n is delta(n) = A055034(n).

Here the smallest nonnegative complete system modulo n is used: {0,1,...,n-1}, and the reduced residue system modulo n (A038566) is the set of numbers k from this set which satisfy gcd(k, n) = 1. The present array lists only the odd numbers. For n = 1 one should take 0 because gcd(0, 1) = 1, but because 1 == 0 (mod 1) we prefer the odd 1.

This is the sub-array obtained from A038566 by deleting the even numbers.

In the multiplicative group Modd n (see a comment in A203571) each of the delta(n) members of row n forms a reduced residue class Modd n with only odd numbers. E.g., n=4 (only the positive members are listed, the negative members should be amended): [1] = {1, 7, 9, 15, 17, 23, 25, 31, 33, 39,...}; [3] = {3, 5, 11, 13, 19, 21, 27, 29, 35, 37...}. Multiplication Modd n can be done class-wise: 7*15 == 1*1 (Modd 4) = 1; 11*13 ==3*3 (Modd 4) = 1; 7*5 == 1*3 (Modd 4) = 3.

The orders 'Moddulo' n of the elements in row n are given in A216320.

Examples

			The array starts:
n\k 1  2   3   4   5   6   7   8   9...
---------------------------------------
1   1
2   1
3   1
4   1  3
5   1  3
6   1  5
7   1  3   5
8   1  3   5   7
9   1  5   7
10  1  3   7   9
11  1  3   5   7   9
12  1  5   7  11
13  1  3   5   7   9  11
14  1  3   5   9  11  13
15  1  7  11  13
16  1  3   5   7   9  11  13  15
17  1  3   5   7   9  11  13  15
18  1  5   7  11  13  17
19  1  3   5   7   9  11  13  15  17
20  1  3   7   9  11  13  17  19
...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..11703 (rows 1 <= n <= 240, flattened)
• Wolfdieter Lang, On the Equivalence of Three Complete Cyclic Systems of Integers, arXiv:2008.04300 [math.NT], 2020.

Crossrefs

Cf. A038566 (row n lists all numbers in the reduced residue system modulo n).

Programs

Mathematica

Table[Select[Range[1, n, 2], GCD[#, n] == 1 &], {n, 20}] (* Michael De Vlieger, Oct 15 2020 *)

PARI

row(n) = select(x->(((x%2)==1) && (gcd(n, x)==1)), [1..n]); \\ Michel Marcus, Jun 10 2020

Formula

a(n, k) is the k-th odd member of the smallest nonnegative reduced residue system modulo n. See the comment above.