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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
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

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.

LINKS

Table of n, a(n) for n=1..107.

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

FORMULA

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

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

Cf. A055034, A072451, A038566, A333856.

Sequence in context: A347499 A136311 A243884 * A126260 A264846 A265691

Adjacent sequences: A334425 A334426 A334427 * A334429 A334430 A334431

KEYWORD

nonn,tabf,easy

AUTHOR

Wolfdieter Lang, Jun 27 2020

STATUS

approved

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Last modified February 6 20:15 EST 2023. Contains 360111 sequences. (Running on oeis4.)