A057593 Triangle T(n, k) giving period length of the periodic sequence k^i (i >= imin) mod n (n >= 2, 1 <= k <= n-1).

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

Offset

2,3

Comments

From Wolfdieter Lang, Sep 04 2017: (Start)

i) If gcd(n, k) = 1 then imin = imin(n, k) = 0 and the length of the period P = T(n, k) = order(n, k), given in A216327 corresponding to the numbers of A038566. This is due to Euler's theorem. E.g., T(4, 3) = 2 because A216327(4, 2) = 2 corresponding to A038566(4, 2) = 3.

ii) If gcd(n, k) is not 1 then the smallest nonnegative index imin = imin(n, k) is obtained from A290601 with the corresponding length of the period given in A290602. Also in this case the sequence always becomes periodic, because one of the possible values from {0, 1, ..., n-1} has to appear a second time because the sequence has more than n entries. Example: T(4, 2) = 1 because imin is given by A290601(1, 1) = 2 (corresponding to the present n = 4, k = 2 values) with the length of the period P given by A290602(1, 1) = 1. (End)

Links

Michael De Vlieger, Table of n, a(n) for n = 2..19901 (rows 2 <= n <= 200).

Example

If n=7, k=2, (imin = 0) the sequence is 1,2,4,1,2,4,1,2,4,... of period 3, so T(7,2) = 3. The triangle T(n, k) begins:
n \ k 1   2   3  4   5   6   7   8  9  10  11  12  13  14 15 16 17 ...
2:    1
3:    1   2
4:    1   1   2
5:    1   4   4  2
6:    1   2   1  1   2
7:    1   3   6  3   6   2
8:    1   1   2  1   2   1   2
9:    1   6   1  3   6   1   3   2
10:   1   4   4  2   1   1   4   4  2
11:   1  10   5  5   5  10  10  10  5   2
12:   1   2   2  1   2   1   2   2  1   1   2
13:   1  12   3  6   4  12  12   4  3   6  12   2
14:   1   3   6  3   6   2   1   1  3   6   3   6   2
15:   1   4   4  2   2   1   4   4  2   1   2   4   4  2
16:   1   1   4  1   4   1   2   1  2   1   4   1   4  1  2
17:   1   8  16  4  16  16  16   8  8  16  16  16   4  16  8  2
18:   1   6   1  3   6   1   3   2  1   1   6   1   3   6  1  1  2
... Reformatted and extended. - Wolfdieter Lang, Sep 04 2017
From Wolfdieter Lang, Sep 04 2017: (Start)
The  table imin(n, k) begins:
n \ k 1   2   3   4   5   6   7   8  9  10  11  12  13  14  15  16 17 ...
2:    0
3:    0   0
4:    0   2   0
5:    0   0   0   0
6:    0   1   1   1   0
7:    0   0   0   0   0   0
8:    0   3   0   2   0   3   0
9:    0   0   2   0   0   2   0   0
10:   0   1   0   1   1   1   0   1  0
11:   0   0   0   0   0   0   0   0  0   0
12:   0   2   1   1   0   2   0   1  1   2   0
13:   0   0   0   0   0   0   0   0  0   0   0   0
14:   0   1   0   1   0   1   1   1  0   1   0   1   0
15:   0   0   1   0   1   1   0   0  1   1   0   1   0   0
16:   0   4   0   2   0   4   0   2  0   4   0   2   0   4   0
17:   0   0   0   0   0   0   0   0  0   0   0   0   0   0   0   0
18:   0   1   2   1   0   2   0   1  1   1   0   2   0   1   2   1  0
... (End)

Mathematica

period[lst_] := Module[{n, i, j}, n=Length[lst]; For[j=2, j <= n, j++, For[i=1, i<j, i++, If[lst[[i]] == lst[[j]], Return[{i-1, j-i}]]]]; Return[{0, 0}]]; T[n_, k_] := Module[{t, p}, t=Table[PowerMod[k, i, n], {i, 0, 2n}]; p=period[t][[2]]; p]; Table[T[n, k], {n, 2, 14}, {k, 1, n - 1}] // Flatten (* Jean-François Alcover, Feb 04 2015 *)

Crossrefs

Cf. A057594, A057595.

Cf. A086145 (prime rows), A216327 (entries with gcd(n,k) = 1), A139366.

Cf. A038566, A216327, A290601, A290602.

Sequence in context: A326376 A106052 A050473 * A117008 A337366 A153917

Adjacent sequences: A057590 A057591 A057592 * A057594 A057595 A057596

Keyword

nonn,tabl,nice

Author

Gottfried Helms, Oct 05 2000

Extensions

Constraint on k changed from 2 <= k <= n to 1 <= k < n, based on comment from Franklin T. Adams-Watters, Jan 19 2006, by David Applegate, Mar 11 2014

Name changed and table extended by Wolfdieter Lang, Sep 04 2017

Status

approved