cp's OEIS Frontend

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).

%I A334428 #13 Jul 02 2020 04:51:54
%S A334428 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,
%T A334428 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,
%U A334428 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
%N 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).
%C A334428 The length of row n is A072451(n) = A055034(2*n-1), for n >= 1.
%C A334428 See the Brändli-Beyne link, and A333856 for the definition and some examples of this mod* system.
%C A334428 This reduced residue system mod* (2*n - 1) will be called RRS*(2*n - 1).
%C A334428 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).
%C A334428 The modular arithmetic is multiplicative but not additive for mod*. See A333856 for examples.
%H A334428 Gerold Brändli and Tim Beyne, <a href="https://arxiv.org/abs/1504.02757">Modified Congruence Modulo n with Half the Amount of Residues</a>, arXiv:1504.02757 [math.NT], 2016.
%F A334428 T(1, 1) = 0, T(n, k) = A038566(2*n - 1, k) for k = 1, 2, ..., A072451(n), for n >= 2.
%e A334428 The irregular triangle T(n, k) begins (b = 2*n - 1):
%e A334428 n    b \k  1 2 3 4 5  6  7  8  9 10 11 12 13 14 15 16 17 18 ...
%e A334428 ---------------------------------------------------------------
%e A334428 1    1:    0
%e A334428 2    3:    1
%e A334428 3    5:    1 2
%e A334428 4    7:    1 2 3
%e A334428 5    9:    1 2 4
%e A334428 6   11:    1 2 3 4 5
%e A334428 7   13:    1 2 3 4 5  6
%e A334428 8   15:    1 2 4 7
%e A334428 9   17:    1 2 3 4 5  6  7  8
%e A334428 10  19:    1 2 3 4 5  6  7  8  9
%e A334428 11  21:    1 2 4 5 8 10
%e A334428 12  23:    1 2 3 4 5  6  7  8  9 10 11
%e A334428 13  25:    1 2 3 4 6  7  8  9 11 12
%e A334428 14  27:    1 2 4 5 7  8 10 11 13
%e A334428 15  29:    1 2 3 4 5  6  7  8  9 10 11 12 13 14
%e A334428 16  31:    1 2 3 4 5  6  7  8  9 10 11 12 13 14 15
%e A334428 17  33:    1 2 4 5 7  8 10 13 14 16
%e A334428 18  35:    1 2 3 4 6  8  9 11 12 13 16 17
%e A334428 19  37:    1 2 3 4 5  6  7  8  9 10 11 12 13 14 15 16 17 18
%e A334428 20  39:    1 2 4 5 7  8 10 11 14 16 17 19
%e A334428 ...
%e A334428 -----------------------------------------------------------
%e A334428 For n = 5 (b = 9) see the example in A333856.
%t A334428 Array[Function[{m, b}, Select[Range[1, m], GCD[#, b] == 1 &] /. {} -> {0}] @@ {# - 1, 2 # - 1} &, 16] // Flatten (* _Michael De Vlieger_, Jun 27 2020 *)
%Y A334428 Cf. A055034, A072451, A038566, A333856.
%K A334428 nonn,tabf,easy
%O A334428 1,4
%A A334428 _Wolfdieter Lang_, Jun 27 2020