A317175 Square array T(n, k) read by antidiagonals upwards, n > 0 and k > 0: T(n, k) is the least m > 0 such that m * n contains k as a substring in its decimal representation.

1, 5, 2, 4, 1, 3, 3, 4, 15, 4, 2, 3, 1, 2, 5, 2, 4, 8, 8, 25, 6, 2, 2, 6, 1, 5, 3, 7, 2, 3, 5, 8, 13, 2, 35, 8, 2, 3, 5, 4, 1, 4, 9, 4, 9, 1, 3, 4, 2, 9, 12, 18, 6, 45, 10, 1, 2, 4, 3, 5, 1, 14, 2, 3, 5, 11, 1, 2, 3, 5, 7, 8, 12, 16, 23, 34, 55, 12, 1, 1, 3, 4

Offset: 1

Author

Rémy Sigrist, Jul 23 2018

Keywords

• nonn
• base
• tabl

Comments

This sequence is well defined: for any n > 0 and k > 0:

- ceiling(k * 10^A055642(n)/n) * n starts with k,

- hence T(n, k) <= ceil(k * 10^A055642(n)/n) <= 10 * k,

- and every column is bounded,

- the conjectured maximum values for the first 9 columns are: 5, 12, 17, 32, 25, 24, 35, 32, 72.

Examples

			Array T(n, k) 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
    2|    5    1   15    2   25    3   35    4   45    5   55    6
    3|    4    4    1    8    5    2    9    6    3   34   37    4
    4|    3    3    8    1   13    4   18    2   23   25   28    3
    5|    2    4    6    8    1   12   14   16   18    2   22   24
    6|    2    2    5    4    9    1   12    3   15   17   19    2
    7|    2    3    5    2    5    8    1    4    7   15   16   16
    8|    2    3    4    3    7    2    9    1   12   13   14   14
    9|    2    3    4    5    5    4    3    2    1   12   13   14
   10|    1    2    3    4    5    6    7    8    9    1   11   12
		

Crossrefs

Cf. A055642, A317173.

Programs

PARI

T(n, k, base=10) = { my (w=base^#digits(k, base)); for (m=1, oo, my (mn=m*n); while (mn >= k, if (mn % w == k, return (m), mn \= base))) }

Formula

T(1, k) = k.

T(n, n) = 1.

T(n, 1) = A317173(n).