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

1,2

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.

Links

Table of n, a(n) for n=1..82.

Formula

T(1, k) = k.

T(n, n) = 1.

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

Example

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

Prog

(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))) }

Crossrefs

Cf. A055642, A317173.

Sequence in context: A267120 A267484 A181697 * A257480 A181696 A157121

Adjacent sequences: A317172 A317173 A317174 * A317176 A317177 A317178

Keyword

nonn,base,tabl

Author

Rémy Sigrist, Jul 23 2018

Status

approved