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A010766
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Triangle read by rows: row n gives the numbers floor(n/k), k = 1..n.
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78
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1, 2, 1, 3, 1, 1, 4, 2, 1, 1, 5, 2, 1, 1, 1, 6, 3, 2, 1, 1, 1, 7, 3, 2, 1, 1, 1, 1, 8, 4, 2, 2, 1, 1, 1, 1, 9, 4, 3, 2, 1, 1, 1, 1, 1, 10, 5, 3, 2, 2, 1, 1, 1, 1, 1, 11, 5, 3, 2, 2, 1, 1, 1, 1, 1, 1, 12, 6, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 13, 6, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1
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OFFSET
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1,2
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COMMENTS
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Number of times k occurs as divisor of numbers not greater than n. - Reinhard Zumkeller, Mar 19 2004
Viewed as a partition, row n is the smallest partition that contains every partition of n in the usual ordering. - Franklin T. Adams-Watters, Mar 11 2006
Row n is the partition whose Young diagram is the union of Young diagrams of all partitions of n (rewording of Franklin T. Adams-Watters's comment). - Harry Richman, Jan 13 2022
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LINKS
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FORMULA
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Let T(n,0) = n+1, then T(n,k) = (sum of the k preceding elements in the previous column) minus (sum of the k preceding elements in same column). - Mats Granvik, Gary W. Adamson, Feb 20 2010
T(n,k) = 1 + T(n-k,k) (where T(n-k,k) = 0 if n < 2*k). - Robert Israel, Sep 01 2014
T(n,k) = T(floor(n/k),1) if k>1; T(n,1) = 1 - Sum_{i=2..n} A008683(i)*T(n,i). If we modify the formula to T(n,1) = 1 - Sum_{i=2..n} A008683(i)*T(n,i)/i^s, where s is a complex variable, then the first column becomes the partial sums of the Riemann zeta function. - Mats Granvik, Apr 27 2016
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EXAMPLE
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Triangle starts:
1: 1;
2: 2, 1;
3: 3, 1, 1;
4: 4, 2, 1, 1;
5: 5, 2, 1, 1, 1;
6: 6, 3, 2, 1, 1, 1;
7: 7, 3, 2, 1, 1, 1, 1;
8: 8, 4, 2, 2, 1, 1, 1, 1;
9: 9, 4, 3, 2, 1, 1, 1, 1, 1;
10: 10, 5, 3, 2, 2, 1, 1, 1, 1, 1;
11: 11, 5, 3, 2, 2, 1, 1, 1, 1, 1, 1;
12: 12, 6, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1;
13: 13, 6, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1;
14: 14, 7, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1;
15: 15, 7, 5, 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1;
16: 16, 8, 5, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1;
17: 17, 8, 5, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1;
18: 18, 9, 6, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1;
19: 19, 9, 6, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
20: 20, 10, 6, 5, 4, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
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MAPLE
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seq(seq(floor(n/k), k=1..n), n=1..20); # Robert Israel, Sep 01 2014
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MATHEMATICA
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Flatten[Table[Floor[n/k], {n, 20}, {k, n}]] (* Harvey P. Dale, Nov 03 2012 *)
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PROG
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(Haskell)
a010766 = div
a010766_row n = a010766_tabl !! (n-1)
a010766_tabl = zipWith (map . div) [1..] a002260_tabl
(PARI) a(n)=t=floor((-1+sqrt(1+8*(n-1)))/2); (t+1)\(n-t*(t+1)/2) \\ Edward Jiang, Sep 10 2014
(PARI) T(n, k) = sum(i=1, n, (i % k) == 0); \\ Michel Marcus, Apr 08 2017
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CROSSREFS
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Finite differences of rows: A075993.
Columns of this triangle:
T(n,1) = n,
Rows of this triangle (with infinite trailing zeros):
...
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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