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A162320
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Array read by antidiagonals: a(n,m) = the number of digits of m when written in base n. The top row is the number of digits for each m in base 2.
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2
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1, 1, 2, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 3, 1, 1, 1, 2, 2, 3, 1, 1, 1, 1, 2, 2, 3, 1, 1, 1, 1, 2, 2, 2, 4, 1, 1, 1, 1, 1, 2, 2, 2, 4, 1, 1, 1, 1, 1, 2, 2, 2, 3, 4, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 4, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 1, 1, 1, 1, 1, 1, 1, 2, 2
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OFFSET
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1,3
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COMMENTS
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A162319 is the same array with the lengths of base 1 numbers in the top row.
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LINKS
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EXAMPLE
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Array read by antidiagonals begins:
1;
1, 2;
1, 1, 2;
1, 1, 2, 3;
1, 1, 1, 2, 3;
1, 1, 1, 2, 2, 3;
1, 1, 1, 1, 2, 2, 3;
1, 1, 1, 1, 2, 2, 2, 4;
1, 1, 1, 1, 1, 2, 2, 2, 4;
...
Array adjusted such that the rows represent base n and the columns m:
m
1 2 3 4 5 6 7 8 9 10
------------------------------
base 2: 1, 2, 2, 3, 3, 3, 3, 4, 4, (4);
base 3: 1, 1, 2, 2, 2, 2, 2, 2, (3, 3);
base 4: 1, 1, 1, 2, 2, 2, 2, (2, 2, 2);
base 5: 1, 1, 1, 1, 2, 2, (2, 2, 2, 2);
base 6: 1, 1, 1, 1, 1, (2, 2, 2, 2, 2);
base 7: 1, 1, 1, 1, (1, 1, 2, 2, 2, 2);
base 8: 1, 1, 1, (1, 1, 1, 1, 2, 2, 2);
base 9: 1, 1, (1, 1, 1, 1, 1, 1, 2, 2);
base 10: 1, (1, 1, 1, 1, 1, 1, 1, 1, 1);
...
For n = 12, a(12) is found in the second position in row 5 in the array read by antidiagonals. This equates to m = 2, base n = 5. The number m = 2 in base n = 5 requires 1 digit, thus a(12) = 1.
For n = 20, a(20) is found in the fifth position in row 6 in the array read by antidiagonals. This equates to m = 5, base n = 3. The number m = 5 in base n = 3 requires 2 digits, thus a(20) = 2. (End)
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MATHEMATICA
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a162320[n_] := Block[{t = {}, i, j}, For[i = 1, i <= n, i++, For[j = i, j > 1, j--, AppendTo[t, Floor@Log[j, i - j + 1] + 1]]]; t]]; a162320[14] (* Michael De Vlieger, Jan 02 2015 *)
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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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