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Irregular triangle of multiplicative orders mod n for the elements of the smallest positive reduced residue system mod n.
5

%I #31 Sep 13 2023 08:54:38

%S 1,1,1,2,1,2,1,4,4,2,1,2,1,3,6,3,6,2,1,2,2,2,1,6,3,6,3,2,1,4,4,2,1,10,

%T 5,5,5,10,10,10,5,2,1,2,2,2,1,12,3,6,4,12,12,4,3,6,12,2,1,6,6,3,3,2,1,

%U 4,2,4,4,2,4,2,1,4,4,2,2,4,4,2

%N Irregular triangle of multiplicative orders mod n for the elements of the smallest positive reduced residue system mod n.

%C The sequence of the row lengths is phi(n) = A000010 (Euler's totient).

%C For the notion 'reduced residue system mod n' which has, as a set, order phi(n) = A000010(n), see e.g., the Apostol reference p. 113. Here such a system with the smallest positive numbers is used. (In the Apostol reference 'order of a modulo n' is called 'exponent of a modulo n'. See the definition on p. 204.)

%C See A038566 where the reduced residue system mod n appears in row n.

%C In the chosen smallest reduced residue system mod n one can replace each element by any congruent mod n one, and the given order modulo n list will, of course, be the same. E.g., n=5, {6, -3, 13, -16} also has the orders modulo 5: 1 4 4 2, respectively.

%C Each order modulo n divides phi(n). See the Niven et al. reference, Corollary 2.32, p. 98.

%C The maximal order modulo n is given in A002322(n).

%C For the analog table of orders Modd n see A216320.

%D T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976.

%D I. Niven, H. S. Zuckerman, and H. L. Montgomery, An Introduction to the Theory of Numbers, Fifth edition, Wiley, 1991.

%H Michael De Vlieger, <a href="/A216327/b216327.txt">Table of n, a(n) for n = 1..12232</a> (rows n = 1..200, flattened.)

%F a(n,k) = order A038566(n,k) modulo n, n >= 1, k=1, 2, ..., phi(n) = A000010(n). This is the order modulo n of the k-th element of the smallest reduced residue system mod n (when their elements are listed increasingly).

%e This irregular triangle begins:

%e n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

%e 1: 1

%e 2: 1

%e 3: 1 2

%e 4: 1 2

%e 5: 1 4 4 2

%e 6: 1 2

%e 7: 1 3 6 3 6 2

%e 8: 1 2 2 2

%e 9: 1 6 3 6 3 2

%e 10: 1 4 4 2

%e 11: 1 10 5 5 5 10 10 10 5 2

%e 12: 1 2 2 2

%e 13: 1 12 3 6 4 12 12 4 3 6 12 2

%e 14: 1 6 6 3 3 2

%e 15: 1 4 2 4 4 2 4 2

%e 16: 1 4 4 2 2 4 4 2

%e 17: 1 8 16 4 16 16 16 8 8 16 16 16 4 16 8 2

%e 18: 1 6 3 6 3 2

%e 19: 1 18 18 9 9 9 3 6 9 18 3 6 18 18 18 9 9 2

%e 20: 1 4 4 2 2 4 4 2

%e ...

%e a(3,2) = 2 because A038566(3,2) = 2 and 2^1 == 2 (mod 3), 2^2 = 4 == 1 (mod 3).

%e a(7,3) = 6 because A038566(7,3) = 3 and 3^1 == 3 (mod 7), 3^2 = 9 == 2 (mod 7), 3^3 = 2*3 == 6 (mod 7), 3^4 == 6*3 == 4 (mod 7), 3^5 == 4*3 == 5 (mod 7) and 3^6 == 5*3 == 1 (mod 7). The notation == means 'congruent'.

%e The maximal order modulo 7 is 6 = A002322(7) = phi(7), and it appears twice: A111725(7) = 2.

%e The maximal order modulo 14 is 6 = A002322(14) = 1*6.

%t Table[Table[MultiplicativeOrder[k,n],{k,Select[Range[n],GCD[#,n]==1&]}],{n,1,13}]//Grid (* _Geoffrey Critzer_, Jan 26 2013 *)

%o (PARI) rowa(n) = select(x->gcd(n, x)==1, [1..n]); \\ A038566

%o row(n) = apply(znorder, apply(x->Mod(x, n), rowa(n))); \\ _Michel Marcus_, Sep 12 2023

%Y Cf. A038566, A002322 (maximal order), A111725 (multiplicity of max order), A216320 (Modd n analog).

%Y Cf. A086145

%K nonn,tabf

%O 1,4

%A _Wolfdieter Lang_, Sep 28 2012