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

Author

Wolfdieter Lang, Aug 30 2017

Keywords

• nonn
• tabf

Comments

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).

Examples

			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)
...
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Crossrefs

Cf. A002808, A290599, A290600, A290601.