A334428 Irregular triangle read by rows: row n gives the members of the smallest nonnegative reduced residue system in the modified congruence modulo 2*n - 1 by Brändli and Beyne, called mod*(2*n - 1).

0, 1, 1, 2, 1, 2, 3, 1, 2, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 1, 2, 4, 7, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 4, 5, 8, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 1, 2, 4, 5, 7, 8, 10, 11, 13, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15

Offset: 1

Author

Wolfdieter Lang, Jun 27 2020

Keywords

• nonn
• tabf
• easy

Comments

The length of row n is A072451(n) = A055034(2*n-1), for n >= 1.

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.

Examples

			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.
		

Links

• Gerold Brändli and Tim Beyne, Modified Congruence Modulo n with Half the Amount of Residues, arXiv:1504.02757 [math.NT], 2016.

Crossrefs

Cf. A055034, A072451, A038566, A333856.

Programs

Mathematica

Array[Function[{m, b}, Select[Range[1, m], GCD[#, b] == 1 &] /. {} -> {0}] @@ {# - 1, 2 # - 1} &, 16] // Flatten (* Michael De Vlieger, Jun 27 2020 *)

Formula

T(1, 1) = 0, T(n, k) = A038566(2*n - 1, k) for k = 1, 2, ..., A072451(n), for n >= 2.