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A216327 Irregular triangle of orders mod n for the elements of the smallest positive reduced residue system mod n. 2
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, 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, 4, 2, 4, 4, 2, 4, 2, 1, 4, 4, 2, 2, 4, 4, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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

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.)

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

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.

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

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

For the analog table of orders Modd n see A216320.

REFERENCES

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

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

LINKS

Table of n, a(n) for n=1..80.

FORMULA

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).

EXAMPLE

This irregular triangle begins:

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

1:  1

2:  1

3:  1  2

4:  1  2

5:  1  4  4  2

6:  1  2

7:  1  3  6  3  6  2

8:  1  2  2  2

9:  1  6  3  6  3  2

10: 1  4  4  2

11: 1 10  5  5  5 10 10 10  5   2

12: 1  2  2  2

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

14: 1  6  6  3  3  2

15: 1  4  2  4  4  2  4  2

16: 1  4  4  2  2  4  4  2

17: 1  8 16  4 16 16 16  8  8  16 16 16   4 16   8  2

18: 1  6  3  6  3  2

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

20: 1  4  4  2  2  4  4  2

...

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

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'.

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

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

MATHEMATICA

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

CROSSREFS

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

Cf. A086145

Sequence in context: A002852 A266081 A188440 * A099875 A079499 A166235

Adjacent sequences:  A216324 A216325 A216326 * A216328 A216329 A216330

KEYWORD

nonn,tabf

AUTHOR

Wolfdieter Lang, Sep 28 2012

STATUS

approved

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Last modified September 30 12:21 EDT 2020. Contains 337439 sequences. (Running on oeis4.)