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

1,3

Links

Table of n, a(n) for n=1..73.

Rémy Sigrist, PARI program for A300653

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.

Example

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

Prog

(PARI) See Links section.

Crossrefs

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

Sequence in context: A303696 A131074 A059268 * A256009 A123937 A138882

Adjacent sequences: A300650 A300651 A300652 * A300654 A300655 A300656

Keyword

nonn,base,tabl

Author

Rémy Sigrist, Mar 10 2018

Status

approved