A010766 Triangle read by rows: row n gives the numbers floor(n/k), k = 1..n.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• tabl
• easy
• nice

Comments

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 sums = A006218. - Gary W. Adamson, Oct 30 2007

A014668 = eigensequence of the triangle. A163313 = A010766 * A014668 (diagonalized) as an infinite lower triangular matrix. - Gary W. Adamson, Jul 30 2009

A018805(T(n,k)) = A242114(n,k). - Reinhard Zumkeller, May 04 2014

Viewed as partitions, all rows are self-conjugate. - Matthew Vandermast, Sep 10 2014

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

Examples

			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;
  ...
		

References

• Florian Cajori, A History of Mathematical Notations, Dover edition (2012), par. 407.

Links

• T. D. Noe, Rows n = 1..50 of triangle, flattened

Crossrefs

Another version of A003988.

Finite differences of rows: A075993.

Cf. related triangles: A002260, A013942, A051731, A163313, A277646, A277647.

Cf. related sequences: A006218, A014668, A115725.

Columns of this triangle:

T(n,1) = n,

T(n,2) = A008619(n-2) for n>1,

T(n,3) = A008620(n-3) for n>2,

T(n,4) = A008621(n-4) for n>3,

T(n,5) = A002266(n) for n>4,

T(n,n) = A000012(n) = 1.

Rows of this triangle (with infinite trailing zeros):

T(1,k) = A000007(k-1),

T(2,k) = A033322(k),

T(3,k) = A278105(k),

T(4,k) = A033324(k),

T(5,k) = A033325(k),

T(6,k) = A033326(k),

T(7,k) = A033327(k),

T(8,k) = A033328(k),

T(9,k) = A033329(k),

T(10,k) = A033330(k),

...

T(99,k) = A033419(k),

T(100,k) = A033420(k),

T(1000,k) = A033421(k),

T(10^4,k) = A033422(k),

T(10^5,k) = A033427(k),

T(10^6,k) = A033426(k),

T(10^7,k) = A033425(k),

T(10^8,k) = A033424(k),

T(10^9,k) = A033423(k).

Programs

Haskell

a010766 = div
a010766_row n = a010766_tabl !! (n-1)
a010766_tabl = zipWith (map . div) [1..] a002260_tabl
-- Reinhard Zumkeller, Apr 29 2015, Aug 13 2013, Apr 13 2012

Maple

seq(seq(floor(n/k),k=1..n),n=1..20); # Robert Israel, Sep 01 2014

Mathematica

Flatten[Table[Floor[n/k],{n,20},{k,n}]] (* Harvey P. Dale, Nov 03 2012 *)

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

Formula

G.f.: 1/(1-x)*Sum_{k>=1} x^k/(1-y*x^k). - Vladeta Jovovic, Feb 05 2004

Triangle A010766 = A000012 * A051731 as infinite lower triangular matrices. - Gary W. Adamson, Oct 30 2007

Equals A000012 * A051731 as infinite lower triangular matrices. - Gary W. Adamson, Nov 14 2007

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) = (n - A048158(n,k)) / k. - Reinhard Zumkeller, Aug 13 2013

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

Extensions

Cross references edited by Jason Kimberley, Nov 23 2016