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A216319 Irregular triangle: row n lists the odd numbers of the reduced residue system modulo n. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

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.

LINKS

Table of n, a(n) for n=1..87.

FORMULA

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

EXAMPLE

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

...

PROG

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

CROSSREFS

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

Sequence in context: A053575 A293485 A250207 * A309425 A218355 A103790

Adjacent sequences:  A216316 A216317 A216318 * A216320 A216321 A216322

KEYWORD

nonn,easy,tabf

AUTHOR

Wolfdieter Lang, Sep 21 2012

STATUS

approved

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Last modified September 26 21:46 EDT 2020. Contains 337377 sequences. (Running on oeis4.)