A290601 -id:A290601 - cp's OEIS frontend

A290602 Irregular triangle read by rows. T(n, k) gives the period length of the periodic sequence {A290600(n, k)^i}_{i >= A290601(n, k)} (mod A002808(n)), for n >= 1 and k = 1..A290599(n).

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

Offset: 1

Wolfdieter Lang, Aug 30 2017

The length of row n is A290599(n).

See A290601 for the proof that this sequence is defined, and the definition of the type of periodicity (imin,P) with imin = A290601(n, k) and the period length P = T(n, k).

			The irregular triangle T(n, k) begins (N(n) = A002808(n)):
n   N(n) \ k  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 ...
1   4         1
2   6         2  1  1
3   8         1  1  1
4   9         1  1
5   10        4  2  1  1  4
6   12        2  2  1  1  2  1  1
7   14        3  3  2  1  1  6  6
8   15        4  2  1  2  1  4
9   16        1  1  1  1  1  1  1
10  18        6  1  3  1  2  1  1  1  6  1  3
11  20        4  2  1  1  4  1  4  2  2  1  4
12  21        6  2  1  3  6  2  1  3
13  22       10  5 10 10  2  1  1  5  5 10  5
14  24        2  2  1  1  2  1  1  1  2  2  1  1  2  2  1
15  25        1  1  1  1
...
T(5, 1) = 4 because A290600(5, 1) = 2, N(5) = A002808(5) = 10, A290601(5, 1) = 1 and {2^i}_{i>=1} (mod 10) == {repeat(2,4,8,6)} with period length 4. This is of the type (1,4).
T(7, 6) = 6 because A290600(7, 6) = 10, N(7) = A002808(7) = 14, A290601(7, 6) = 1 and {10^i}_{i>=1} (mod 14) == {repeat(10, 2, 6, 4, 12, 8)} with period length 4. Type (1,6).
The sequence {A290600(10, 1)^i}_{i >= A290601(10, 1)} (mod A002808(10)) = {2^i}_{i >= 1} (mod 18) is periodic with period length P = T(10, 1) = 6. Namely, {repeat(2, 4, 8, 16, 14, 10)}, of type (1,6).
The periodicity types (imin,P) = (A290601(n, k), A290602(n, k)) begin:
n   N(n) \ k    1     2      3      4     5     6     7     8     9      10    11
1   4         (2,1)
2   6         (1,2) (1,1)  (1,1)
3   8         (3,1) (2,1)  (3,1)
4   9         (2,1) (2,1)
5   10        (1,4) (1,2)  (1,1)  (1,1) (1,4)
6   12        (2,2) (1,2)  (1,1)  (2,1) (1,2) (1,1) (2,1)
7   14        (1,3) (1,3)  (1,2)  (1,1) (1,1) (1,6) (1,6)
8   15        (1,4) (1,2)  (1,1)  (1,2) (1,1) (1,4)
9   16        (4,1) (2,1)  (4,1)  (2,1) (4,1) (2,1) (4,1)
10  18        (1,6) (2,1)  (1,3)  (2,1) (1,2) (1,1) (1,1) (2,1) (1,6)  (2,1) (1,3)
11  20        (2,4) (1,2)  (1,1)  (2,1) (1,4) (2,1) (1,4) (2,2) (1,2)  (1,1) (2,4)
12  21        (1,6) (1,2)  (1,1)  (1,3) (1,6) (1,2) (1,1) (1,3)
13  22       (1,10) (1,5) (1,10) (1,10) (1,2) (1,1) (1,1) (1,5) (1,5) (1,10) (1,5)
...
----------------------------------------------------------------------------------

Cf. A002808, A290599, A290600, A290601.

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

Original entry on oeis.org

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

Gottfried Helms, Oct 05 2000

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)

			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)

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

Cf. A057594, A057595.
Cf. A086145 (prime rows), A216327 (entries with gcd(n,k) = 1), A139366.
Cf. A038566, A216327, A290601, A290602.

period[lst_] := Module[{n, i, j}, n=Length[lst]; For[j=2, j <= n, j++, For[i=1, iJean-François Alcover, Feb 04 2015 *)

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

A290600 Irregular triangle T(n, k) read by rows: positive numbers non-coprime to A002808(n) and smaller than A002808(n), sorted increasingly.

Original entry on oeis.org

2, 2, 3, 4, 2, 4, 6, 3, 6, 2, 4, 5, 6, 8, 2, 3, 4, 6, 8, 9, 10, 2, 4, 6, 7, 8, 10, 12, 3, 5, 6, 9, 10, 12, 2, 4, 6, 8, 10, 12, 14, 2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 2, 4, 5, 6, 8, 10, 12, 14, 15, 16, 18, 3, 6, 7, 9, 12, 14, 15, 18

Offset: 1

Wolfdieter Lang, Aug 30 2017

The length of row n is A290599(n).

Row n gives the complement of row A038566(A002808(n), k) with respect to [1, 2, ..., A002808(n) - 1].

			The irregular triangle T(n, k) begins (N(n) = A002808(n)):
  n   N(n) \ k  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 ...
  1   4         2
  2   6         2  3  4
  3   8         2  4  6
  4   9         3  6
  5   10        2  4  5  6  8
  6   12        2  3  4  6  8  9 10
  7   14        2  4  6  7  8 10 12
  8   15        3  5  6  9 10 12
  9   16        2  4  6  8 10 12 14
  10  18        2  3  4  6  8  9 10 12 14 15 16
  11  20        2  4  5  6  8 10 12 14 15 16 18
  12  21        3  6  7  9 12 14 15 18
  13  22        2  4  6  8 10 11 12 14 16 18 20
  14  24        2  3  4  6  8  9 10 12 14 15 16 18 20 21 22
  15  25        5 10 15 20
  ...

Cf. A002808, A038566, A290599, A290601, A290602.

Table[With[{c = FixedPoint[n + PrimePi@ # + 1 &, n + PrimePi@ n + 1]}, Select[Range[c - 1], ! CoprimeQ[#, c] &]], {n, 12}] // Flatten (* Michael De Vlieger, Sep 03 2017 *)

T(n, k) = k-th entry in the list of increasingly sorted numbers of the set {m = 1..A002808(n)-1: gcd(n, m) not equal to 1}.

cp's OEIS Frontend

A290602 Irregular triangle read by rows. T(n, k) gives the period length of the periodic sequence {A290600(n, k)^i}_{i >= A290601(n, k)} (mod A002808(n)), for n >= 1 and k = 1..A290599(n).

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

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A290600 Irregular triangle T(n, k) read by rows: positive numbers non-coprime to A002808(n) and smaller than A002808(n), sorted increasingly.

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