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A317175
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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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2
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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FORMULA
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T(1, k) = k.
T(n, n) = 1.
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EXAMPLE
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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
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PROG
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(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))) }
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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