A033426 -id:A033426 - cp's OEIS frontend

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

N. J. A. Sloane

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

			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;
  ...

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

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

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).

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

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

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

a(n)=t=floor((-1+sqrt(1+8*(n-1)))/2);(t+1)\(n-t*(t+1)/2) \\ Edward Jiang, Sep 10 2014

T(n, k) = sum(i=1, n, (i % k) == 0); \\ Michel Marcus, Apr 08 2017

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

Cross references edited by Jason Kimberley, Nov 23 2016

A033420 a(n) = floor(100/n).

Original entry on oeis.org

100, 50, 33, 25, 20, 16, 14, 12, 11, 10, 9, 8, 7, 7, 6, 6, 5, 5, 5, 5, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Jeff Burch

a(n) is also the percentage of the first n elements of the list represented by n. e.g., 2 is 50% of the sequence 1,2. - Dominick Cancilla, Aug 09 2010

Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A033421, A033422, A033423, A033424, A033425, A033426, A057072.

[Floor(100/n): n in [1..100]]; // Wesley Ivan Hurt, Apr 04 2023

Floor[100/Range[100]] (* Harvey P. Dale, May 27 2021 *)

a(n) = 100\n; \\ Seiichi Manyama, Aug 12 2023

A033422 a(n) = floor(10000/n).

Original entry on oeis.org

10000, 5000, 3333, 2500, 2000, 1666, 1428, 1250, 1111, 1000, 909, 833, 769, 714, 666, 625, 588, 555, 526, 500, 476, 454, 434, 416, 400, 384, 370, 357, 344, 333, 322, 312, 303, 294, 285, 277, 270, 263, 256, 250, 243, 238, 232, 227, 222

Offset: 1

Jeff Burch

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A033420, A033421, A033423, A033424, A033425, A033426, A057072.

[Floor(10000/n): n in [1..60]]; // Wesley Ivan Hurt, Apr 04 2023

Floor[10000/Range[100]] (* Wesley Ivan Hurt, Mar 18 2023 *)

a(n) = 10000\n; \\ Seiichi Manyama, Aug 12 2023

A033424 a(n) = floor(10^8/n).

Original entry on oeis.org

100000000, 50000000, 33333333, 25000000, 20000000, 16666666, 14285714, 12500000, 11111111, 10000000, 9090909, 8333333, 7692307, 7142857, 6666666, 6250000, 5882352, 5555555, 5263157, 5000000, 4761904, 4545454, 4347826

Offset: 1

Jeff Burch

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A033420, A033421, A033422, A033423, A033425, A033426, A057072.

[Floor(10^8/n): n in [1..40]]; // Wesley Ivan Hurt, Apr 04 2023

Floor[10^8/Range[60]] (* Wesley Ivan Hurt, Mar 18 2023 *)

a(n) = 10^8\n; \\ Seiichi Manyama, Aug 12 2023

More terms from James Sellers, Jun 20 2000

A033421 a(n) = floor(1000/n).

Original entry on oeis.org

1000, 500, 333, 250, 200, 166, 142, 125, 111, 100, 90, 83, 76, 71, 66, 62, 58, 55, 52, 50, 47, 45, 43, 41, 40, 38, 37, 35, 34, 33, 32, 31, 30, 29, 28, 27, 27, 26, 25, 25, 24, 23, 23, 22, 22, 21, 21, 20, 20, 20, 19, 19, 18, 18, 18, 17, 17, 17, 16, 16, 16, 16

Offset: 1

Jeff Burch

Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A033420, A033422, A033423, A033424, A033425, A033426, A057072.

[Floor(1000/n): n in [1..80]]; // Wesley Ivan Hurt, Apr 04 2023

Floor[1000/Range[70]] (* Harvey P. Dale, Aug 05 2014 *)

a(n) = 1000\n; \\ Seiichi Manyama, Aug 12 2023

Corrected and extended by Harvey P. Dale, Aug 05 2014

A033423 a(n) = floor(10^9/n).

Original entry on oeis.org

1000000000, 500000000, 333333333, 250000000, 200000000, 166666666, 142857142, 125000000, 111111111, 100000000, 90909090, 83333333, 76923076, 71428571, 66666666, 62500000, 58823529, 55555555, 52631578, 50000000

Offset: 1

Jeff Burch

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A033420, A033421, A033422, A033424, A033425, A033426, A057072.

[Floor(10^9/n): n in [1..40]]; // Wesley Ivan Hurt, Apr 04 2023

Floor[10^9/Range[50]] (* Wesley Ivan Hurt, Mar 18 2023 *)

a(n) = 10^9\n; \\ Seiichi Manyama, Aug 12 2023

A033425 a(n) = floor(10^7/n).

Original entry on oeis.org

10000000, 5000000, 3333333, 2500000, 2000000, 1666666, 1428571, 1250000, 1111111, 1000000, 909090, 833333, 769230, 714285, 666666, 625000, 588235, 555555, 526315, 500000, 476190, 454545, 434782, 416666, 400000, 384615

Offset: 1

Jeff Burch

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A033420, A033421, A033422, A033423, A033424, A033426, A033427, A057072.

[Floor(10^7/n): n in [1..40]]; // Wesley Ivan Hurt, Apr 04 2023

Floor[10^7/Range[50]] (* Wesley Ivan Hurt, Mar 18 2023 *)

a(n) = 10^7\n; \\ Seiichi Manyama, Aug 13 2023

A057072 a(n) = floor(10^10/n).

Original entry on oeis.org

10000000000, 5000000000, 3333333333, 2500000000, 2000000000, 1666666666, 1428571428, 1250000000, 1111111111, 1000000000, 909090909, 833333333, 769230769, 714285714, 666666666, 625000000, 588235294, 555555555, 526315789

Offset: 1

Henry Bottomley, Jul 31 2000

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A033330, A033420-A033426, A000007, A033324, A033347, A057066-A057075.

a(n) = 10^10\n; \\ Seiichi Manyama, Aug 12 2023

A277646 Triangle T(n,k) = floor(n^2/k) for 1 <= k <= n^2, read by rows.

Original entry on oeis.org

1, 4, 2, 1, 1, 9, 4, 3, 2, 1, 1, 1, 1, 1, 16, 8, 5, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 25, 12, 8, 6, 5, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 36, 18, 12, 9, 7, 6, 5, 4, 4, 3, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 49, 24, 16, 12, 9, 8, 7, 6

Offset: 1

Jason Kimberley, Nov 09 2016

			The first five rows of the triangle are:
1;
4, 2, 1, 1;
9, 4, 3, 2, 1, 1, 1, 1, 1;
16, 8, 5, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1;
25, 12, 8, 6, 5, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;

Jason Kimberley, Table of n, a(n) for n = 1..10416 (the first 31 rows of the triangle)

Cf. Related triangles: A010766, A277647, A277648.
Rows of this triangle (with infinite trailing zeros):
T(1,k) = A000007(k-1),
T(2,k) = A033324(k),
T(3,k) = A033329(k),
T(4,k) = A033336(k),
T(5,k) = A033345(k),
T(6,k) = A033356(k),
T(7,k) = A033369(k),
T(8,k) = A033384(k),
T(9,k) = A033401(k),
T(10,k) = A033420(k),
T(100,k) = A033422(k),
T(10^3,k) = A033426(k),
T(10^4,k) = A033424(k).
Columns of this triangle:
T(n,1) = A000290(n),
T(n,2) = A007590(n),
T(n,3) = A000212(n),
T(n,4) = A002620(n),
T(n,5) = A118015(n),
T(n,6) = A056827(n),
T(n,7) = A056834(n),
T(n,8) = A130519(n+1),
T(n,9) = A056838(n),
T(n,10)= A056865(n),
T(n,12)= A174709(n+2).

A277646:=func;
[A277646(n,k):k in[1..n^2],n in[1..7]];

Table[Floor[n^2/k], {n, 7}, {k, n^2}] // Flatten (* Michael De Vlieger, Nov 24 2016 *)

T(n,k) = A010766(n^2,k).

A033427 a(n) = floor(10^5/n).

Original entry on oeis.org

100000, 50000, 33333, 25000, 20000, 16666, 14285, 12500, 11111, 10000, 9090, 8333, 7692, 7142, 6666, 6250, 5882, 5555, 5263, 5000, 4761, 4545, 4347, 4166, 4000, 3846, 3703, 3571, 3448, 3333, 3225, 3125, 3030, 2941, 2857, 2777, 2702, 2631

Offset: 1

Jeff Burch

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A033420, A033421, A033422, A033423, A033424, A033425, A033426, A057072.

[Floor(10^5/n): n in [1..60]]; // Wesley Ivan Hurt, Apr 04 2023

Floor[10^5/Range[40]] (* Harvey P. Dale, Apr 29 2013 *)

a(n) = 10^5\n; \\ Seiichi Manyama, Aug 13 2023

cp's OEIS Frontend

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

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A033420 a(n) = floor(100/n).

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A033422 a(n) = floor(10000/n).

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A033424 a(n) = floor(10^8/n).

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A033421 a(n) = floor(1000/n).

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A033423 a(n) = floor(10^9/n).

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A033425 a(n) = floor(10^7/n).

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