A300691 Square array T(n, k) (n >= 1, k >= 1) read by antidiagonals upwards: T(n, k) is the k-th positive number, say m, such that the binary representation of n appears as a substring in the binary representation of 1/m (ignoring the radix point and adding trailing zeros if necessary in case of a terminating expansion).

1, 1, 2, 5, 2, 3, 1, 9, 3, 4, 3, 2, 10, 4, 5, 5, 6, 4, 11, 5, 6, 9, 9, 11, 5, 13, 6, 7, 1, 11, 10, 12, 7, 17, 7, 8, 5, 2, 13, 11, 13, 8, 18, 8, 9, 3, 7, 4, 17, 13, 19, 9, 19, 9, 10, 11, 6, 10, 8, 18, 17, 22, 10, 20, 10, 11, 5, 13, 11, 13, 9, 19, 18, 23, 11, 21

Offset: 1

Author

Rémy Sigrist, Mar 11 2018

Keywords

• nonn
• base
• tabl

Comments

If m appears in the n-th row, then 2*m also appears in the n-th row.

This array has connections with A300653: here n appears in 1/T(n, k), there T(n, k) appears in 1/n.

Examples

			Square array begins:
  n\k|   1   2   3   4   5   6   7   8   9  10  11  12
  ---+------------------------------------------------
    1|   1   2   3   4   5   6   7   8   9  10  11  12  -->  A000027
    2|   1   2   3   4   5   6   7   8   9  10  11  12  -->  A000027
    3|   5   9  10  11  13  17  18  19  20  21  22  23  -->  A300669
    4|   1   2   4   5   7   8   9  10  11  13  14  15
    5|   3   6  11  12  13  19  22  23  24  25  26  27
    6|   5   9  10  11  13  17  18  19  20  21  22  23
    7|   9  11  13  17  18  19  22  25  26  27  29  33
    8|   1   2   4   8   9  11  13  15  16  17  18  19
    9|   5   7  10  13  14  19  20  23  26  27  28  29
   10|   3   6  11  12  19  22  24  25  27  29  35  37
   11|  11  13  19  22  23  25  26  27  29  37  38  43
   12|   5   9  10  13  17  18  19  20  21  23  25  26
		

Links

• Rémy Sigrist, PARI program for A300691

Crossrefs

Cf. A000079, A000225, A153055, A300428, A300653, A300669.

Programs

PARI

See Links section.

Formula

T(1, k) = k.

T(2, k) = k.

T(3, k) = A300669(k).

T(n, 1) = A300428(n).

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

T(A000225(i), k) = T(2*A000225(i), k) for any i > 0.