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%I #33 Oct 15 2020 16:35:47
%S 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,
%T 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,
%U 15,1,5,7,11,13,17,1,3,5,7,9,11,13,15,17,1,3,7,9,11,13,17,19
%N Irregular triangle: row n lists the odd numbers of the reduced residue system modulo n.
%C The length of row n is delta(n) = A055034(n).
%C 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.
%C This is the sub-array obtained from A038566 by deleting the even numbers.
%C 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.
%C The orders 'Moddulo' n of the elements in row n are given in A216320.
%H Michael De Vlieger, <a href="/A216319/b216319.txt">Table of n, a(n) for n = 1..11703</a> (rows 1 <= n <= 240, flattened)
%H Wolfdieter Lang, <a href="https://arxiv.org/abs/2008.04300">On the Equivalence of Three Complete Cyclic Systems of Integers</a>, arXiv:2008.04300 [math.NT], 2020.
%F a(n, k) is the k-th odd member of the smallest nonnegative reduced residue system modulo n. See the comment above.
%e The array starts:
%e n\k 1 2 3 4 5 6 7 8 9...
%e ---------------------------------------
%e 1 1
%e 2 1
%e 3 1
%e 4 1 3
%e 5 1 3
%e 6 1 5
%e 7 1 3 5
%e 8 1 3 5 7
%e 9 1 5 7
%e 10 1 3 7 9
%e 11 1 3 5 7 9
%e 12 1 5 7 11
%e 13 1 3 5 7 9 11
%e 14 1 3 5 9 11 13
%e 15 1 7 11 13
%e 16 1 3 5 7 9 11 13 15
%e 17 1 3 5 7 9 11 13 15
%e 18 1 5 7 11 13 17
%e 19 1 3 5 7 9 11 13 15 17
%e 20 1 3 7 9 11 13 17 19
%e ...
%t Table[Select[Range[1, n, 2], GCD[#, n] == 1 &], {n, 20}] (* _Michael De Vlieger_, Oct 15 2020 *)
%o (PARI) row(n) = select(x->(((x%2)==1) && (gcd(n, x)==1)), [1..n]); \\ _Michel Marcus_, Jun 10 2020
%Y Cf. A038566 (row n lists all numbers in the reduced residue system modulo n).
%K nonn,easy,tabf
%O 1,5
%A _Wolfdieter Lang_, Sep 21 2012
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