A300653 Square array T(n, k) (n >= 1, k >= 1) read by antidiagonals upwards: T(n, k) is the k-th positive number whose binary representation appears as a substring in the binary representation of 1/n (ignoring the radix point and adding trailing zeros if necessary in case of a terminating expansion).

1, 1, 2, 1, 2, 4, 1, 2, 4, 8, 1, 2, 5, 8, 16, 1, 2, 4, 10, 16, 32, 1, 2, 3, 8, 21, 32, 64, 1, 2, 5, 4, 16, 42, 64, 128, 1, 2, 4, 10, 6, 32, 85, 128, 256, 1, 2, 4, 9, 21, 9, 64, 170, 256, 512, 1, 2, 3, 8, 18, 42, 12, 128, 341, 512, 1024, 1, 2, 3, 4, 16, 36, 85

Offset: 1

Author

Rémy Sigrist, Mar 10 2018

Keywords

• nonn
• base
• tabl

Examples

			Square array begins:
  n\k|    1    2    3    4    5    6    7    8    9   10
  ---+--------------------------------------------------
    1|    1    2    4    8   16   32   64  128  256  512  -->  A000079
    2|    1    2    4    8   16   32   64  128  256  512
    3|    1    2    5   10   21   42   85  170  341  682  -->  A000975
    4|    1    2    4    8   16   32   64  128  256  512
    5|    1    2    3    4    6    9   12   19   25   38
    6|    1    2    5   10   21   42   85  170  341  682
    7|    1    2    4    9   18   36   73  146  292  585  -->  A033138
    8|    1    2    4    8   16   32   64  128  256  512
    9|    1    2    3    4    6    7    8   12   14   17
   10|    1    2    3    4    6    9   12   19   25   38
		

Links

• Rémy Sigrist, PARI program for A300653

Crossrefs

Cf. A000079, A000975, A033138, A153055, A300630, A300654.

Programs

PARI

See Links section.

Formula

T(n, 1) = 1.

T(n, 2) = 2.

T(n, 3) > 3 iff n belongs to A300630.

T(2*n, k) = T(n, k).

T(1, k) = A000079(k-1).

T(3, k) = A000975(k).

T(7, k) = A033138(k).

T(n, k) = k iff 1 <= k <= A300654(n).

T(n, k) = n for some k iff n belongs to A000079 or to A153055.