OFFSET
1,4
COMMENTS
See the Brändli-Beyne link, and A333856 for the definition and some examples of this mod* system.
This reduced residue system mod* (2*n - 1) will be called RRS*(2*n - 1).
Compare this table with the one for the reduced residue system modulo 2*n - 1 (called RRS(2*n - 1) = A038566(2*n - 1), but with A038566(1) = 0). For n >= 2 RRS*(2*n-1) consists of the first half of the entries of RRS(2*n - 1).
The modular arithmetic is multiplicative but not additive for mod*. See A333856 for examples.
LINKS
Gerold Brändli and Tim Beyne, Modified Congruence Modulo n with Half the Amount of Residues, arXiv:1504.02757 [math.NT], 2016.
EXAMPLE
The irregular triangle T(n, k) begins (b = 2*n - 1):
n b \k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 ...
---------------------------------------------------------------
1 1: 0
2 3: 1
3 5: 1 2
4 7: 1 2 3
5 9: 1 2 4
6 11: 1 2 3 4 5
7 13: 1 2 3 4 5 6
8 15: 1 2 4 7
9 17: 1 2 3 4 5 6 7 8
10 19: 1 2 3 4 5 6 7 8 9
11 21: 1 2 4 5 8 10
12 23: 1 2 3 4 5 6 7 8 9 10 11
13 25: 1 2 3 4 6 7 8 9 11 12
14 27: 1 2 4 5 7 8 10 11 13
15 29: 1 2 3 4 5 6 7 8 9 10 11 12 13 14
16 31: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
17 33: 1 2 4 5 7 8 10 13 14 16
18 35: 1 2 3 4 6 8 9 11 12 13 16 17
19 37: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
20 39: 1 2 4 5 7 8 10 11 14 16 17 19
...
-----------------------------------------------------------
For n = 5 (b = 9) see the example in A333856.
MATHEMATICA
Array[Function[{m, b}, Select[Range[1, m], GCD[#, b] == 1 &] /. {} -> {0}] @@ {# - 1, 2 # - 1} &, 16] // Flatten (* Michael De Vlieger, Jun 27 2020 *)
CROSSREFS
KEYWORD
nonn,tabf,easy
AUTHOR
Wolfdieter Lang, Jun 27 2020
STATUS
approved