A317173 -id:A317173 - cp's OEIS frontend

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

Rémy Sigrist, Jul 23 2018

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.

			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

Cf. A055642, A317173.

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))) }

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

A317180 a(n) is the least positive multiple of n that contains at least one digit 1 in its decimal representation.

Original entry on oeis.org

1, 10, 12, 12, 10, 12, 14, 16, 18, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 100, 21, 110, 115, 120, 100, 104, 81, 112, 116, 120, 31, 128, 132, 102, 105, 108, 111, 114, 117, 120, 41, 126, 129, 132, 135, 138, 141, 144, 147, 100, 51, 104, 106, 108, 110, 112, 114

Offset: 1

Rémy Sigrist, Jul 23 2018

			The multiples of 3 are: 3, 6, 9, 12, 15, etc.; 12 is the first one containing the digit 1, hence a(3) = 12.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A011531, A187285, A317173.

on1[n_]:=Module[{k=1},While[DigitCount[k*n,10,1]<1,k++];k*n]; Array[on1,60] (* Harvey P. Dale, Apr 09 2022 *)

a(n) = forstep (m=n, oo, n, if (setsearch(Set(digits(m)), 1), return (m)))

a(n) = n * A317173(n).

a(n) <= A187285(n).

Definition clarified by Harvey P. Dale, Apr 09 2022

A319542 Record values in A039932.

Original entry on oeis.org

1, 51, 531, 2571, 15703, 90271, 102053, 530102, 4550102, 4570102, 4580102, 22900501, 134003006, 1002003005, 5001002003, 5003001002, 30005001002, 30005002001, 200030005001, 1000200030005, 5000100020003, 5000300010002

Offset: 1

David Radcliffe, Sep 22 2018

The positive integer n belongs to the sequence iff there exists a positive integer k so that each of the first k multiples of n contains the decimal digit 1, but no smaller n has this property for the same value of k.

			51 is a term because the first three multiples of 51 (51, 102, 153) all contain the digit 1, and 51 is the least positive integer with this property.

Cf. A319548, A039932, A317173. Subset of A011531.

A381530 a(n) is the least k > 0 such that n / k contains a digit 1 in its decimal representation.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 4, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 2, 3, 3, 3, 3, 3, 3, 3, 1, 3, 3, 3, 3, 3, 3, 3, 3, 4, 1, 2, 2, 3, 3, 4, 4, 4, 4, 4, 1, 4, 4, 4, 4, 4, 4, 4, 4, 5, 1, 2, 2, 4, 4, 4, 4, 5, 5, 5, 1, 5, 3, 3, 3, 5, 5, 5, 5, 1

Offset: 1

Ctibor O. Zizka, Feb 26 2025

			n = 7:
7/1 = 7
7/2 = 3.5
7/3 = 2.33...
7/4 = 1.75 contains a digit 1, thus a(7) = 4.

Cf. A317173, A371706.

a[n_] := Module[{k = 1}, While[FreeQ[RealDigits[n/k][[1]], 1], k++]; k]; Array[a, 100] (* Amiram Eldar, Feb 26 2025 *)

cp's OEIS Frontend

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.

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A317180 a(n) is the least positive multiple of n that contains at least one digit 1 in its decimal representation.

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A319542 Record values in A039932.

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A381530 a(n) is the least k > 0 such that n / k contains a digit 1 in its decimal representation.

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