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A216319 Irregular triangle: row n lists the odd numbers of the reduced residue system modulo n. 6
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
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.
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
...
MATHEMATICA
Table[Select[Range[1, n, 2], GCD[#, n] == 1 &], {n, 20}] (* Michael De Vlieger, Oct 15 2020 *)
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 * A355001 A337713 A309425
KEYWORD
nonn,easy,tabf
AUTHOR
Wolfdieter Lang, Sep 21 2012
STATUS
approved

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Last modified April 23 08:14 EDT 2024. Contains 371905 sequences. (Running on oeis4.)