A066799 Square array read by antidiagonals of eventual period of powers of k mod n; period of repeating digits of 1/n in base k.

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

Offset

1,9

Comments

The determinant of the n X n matrix made from the northwest corner of this array is 0^(n-1). - Iain Fox, Mar 12 2018

Links

Iain Fox, Antidiagonals n = 1..141 of array, flattened

Formula

T(n, k) = T(n, k-n) if k > n.

T(n, n) = T(n, n+1) = 1.

T(n, n-1) = 2.

Example

Rows start: 1,1,1,1,1,...; 1,1,1,1,1,...; 1,2,1,1,2,...; 1,1,2,1,1; 1,4,4,2,1,... T(3,2)=2 since the powers of 2 become 1,2,1,2,1,2,... mod 3 with period 2. T(4,2)=1 since the powers of 2 become 1,2,0,0,0,0,... mod 4 with eventual period 1.
Beginning of array:
+-----+--------------------------------------------------------------------
| n\k |  1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  ...
+-----+--------------------------------------------------------------------
|  1  |  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
|  2  |  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
|  3  |  1,  2,  1,  1,  2,  1,  1,  2,  1,  1,  2,  1,  1,  2,  1,  1, ...
|  4  |  1,  1,  2,  1,  1,  1,  2,  1,  1,  1,  2,  1,  1,  1,  2,  1, ...
|  5  |  1,  4,  4,  2,  1,  1,  4,  4,  2,  1,  1,  4,  4,  2,  1,  1, ...
|  6  |  1,  2,  1,  1,  2,  1,  1,  2,  1,  1,  2,  1,  1,  2,  1,  1, ...
|  7  |  1,  3,  6,  3,  6,  2,  1,  1,  3,  6,  3,  6,  2,  1,  1,  3, ...
|  8  |  1,  1,  2,  1,  2,  1,  2,  1,  1,  1,  2,  1,  2,  1,  2,  1, ...
| ... |

Mathematica

t[n_, k_] := For[p = PowerMod[k, n, n]; m = n + 1, True, m++, If[PowerMod[k, m, n] == p, Return[m - n]]]; Flatten[Table[t[n - k + 1, k], {n, 1, 14}, {k, n, 1, -1}]] (* Jean-François Alcover, Jun 04 2012 *)

Prog

(PARI) a(n, k) = my(p=k^n%n); for(m=n+1, +oo, if(k^m%n==p, return(m-n))) \\ Iain Fox, Mar 12 2018

Crossrefs

Columns are A000012, A007733, A007734, A007735, A007736, A007737, A007738, A007739, A007740, A007732. A002322 is the highest value in each row and the least common multiple of each row, while the number of distinct values in each row is A066800.

Sequence in context: A193582 A091887 A144871 * A238900 A037832 A170977

Adjacent sequences: A066796 A066797 A066798 * A066800 A066801 A066802

Keyword

nonn,tabl,base

Author

Henry Bottomley, Dec 20 2001

Status

approved