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A273620
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Table read by ascending antidiagonals: T(n, k) = floor(sqrt(k) * floor(n/sqrt(k) + 1)), for n >= 1, k >= 1.
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2
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2, 3, 1, 4, 2, 1, 5, 4, 3, 2, 6, 4, 3, 4, 2, 7, 5, 5, 4, 2, 2, 8, 7, 5, 6, 4, 2, 2, 9, 7, 6, 6, 4, 4, 2, 2, 10, 8, 8, 8, 6, 4, 5, 2, 3, 11, 9, 8, 8, 6, 7, 5, 5, 3, 3, 12, 11, 10, 10, 8, 7, 5, 5, 6, 3, 3, 13, 11, 10, 10, 8, 7, 7, 5, 6, 3, 3, 3, 14, 12, 12, 12
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OFFSET
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1,1
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COMMENTS
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A261865(n) gives the least k such that T(n, k) = n.
T(n, k) is the floor of the least multiple of sqrt(k) that is greater than n.
T(n, k^2) is a multiple of k.
For squarefree k > 1, T(n,k) = n if and only if n appears in column k.
A327952(n) is the number of appearances of n in row n.
(End)
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LINKS
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FORMULA
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T(n, 1) = n + 1.
T(n, k) = floor(sqrt(k) * floor(n/sqrt(k) + 1)). - Peter Kagey, Apr 07 2020
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EXAMPLE
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A261865(1) = T(1, 1) = floor(sqrt(1) * floor(1/sqrt(1) + 1)) = 2
A261865(2) = T(2, 1) = floor(sqrt(1) * floor(1/sqrt(2) + 1)) = 1
A261865(3) = T(1, 2) = floor(sqrt(2) * floor(2/sqrt(1) + 1)) = 4
Table begins:
n\k | 1 2 3 4 5 6 7 8 9 10
----+------------------------------
1| 2 1 1 2 2 2 2 2 3 3
2| 3 2 3 4 2 2 2 2 3 3
3| 4 4 3 4 4 4 5 5 6 3
4| 5 4 5 6 4 4 5 5 6 6
5| 6 5 5 6 6 7 5 5 6 6
6| 7 7 6 8 6 7 7 8 9 6
7| 8 7 8 8 8 7 7 8 9 9
8| 9 8 8 10 8 9 10 8 9 9
9| 10 9 10 10 11 9 10 11 12 9
10| 11 11 10 12 11 12 10 11 12 12
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MATHEMATICA
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Table[Function[j, Floor[Sqrt@ k Floor[j/Sqrt@ k + 1]]][n - k + 1], {n, 13}, {k, n}] // Flatten (* Michael De Vlieger, May 27 2016 *)
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PROG
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(Haskell)
a273620T :: Integral a => a -> a -> a
a273620T n k = floor $ sqrt k' * c where
(n', k') = (fromIntegral n, fromIntegral k)
c = fromIntegral $ floor $ n' / sqrt k' + 1
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CROSSREFS
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The first column consists of entries in A001951, the second column of entries in A022838, the fourth of entries in A022839, and the fifth of entries in A022840.
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KEYWORD
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AUTHOR
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STATUS
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approved
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