A277646 -id:A277646 - 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

A277647 Triangle T(n,k) = floor(n/sqrt(k)) for 1 <= k <= n^2, read by rows.

Original entry on oeis.org

1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 4, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 4, 3, 3, 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, 7, 4, 4, 3, 3, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Jason Kimberley, Nov 09 2016

			The first five rows of the triangle are:
1;
2, 1, 1, 1;
3, 2, 1, 1, 1, 1, 1, 1, 1;
4, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
5, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 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. A010766, A277646, A277648.

The 1000th row is A033432.

A277647:=func;
[A277647(n,k):k in[1..n^2],n in[1..7]];

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

row(n) = for(k=1, n^2, print1(floor(n/sqrt(k)), ", ")); print("")
trianglerows(n) = for(k=1, n, row(k))
/* Print initial five rows of triangle as follows: */
trianglerows(5) \\ Felix Fröhlich, Nov 12 2016

T(n,k) = A000196(A277646(n,k)).

T(n,k)sqrt(k) <= n < (T(n,k)+1)sqrt(k).

A277648 Triangle T(n,k) = A277647(n, A005117(k)), read by rows.

Original entry on oeis.org

1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 4, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 5, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 4, 4, 3, 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, 8, 5, 4, 3, 3, 3, 2, 2, 2, 2, 2, 1

Offset: 1

Jason Kimberley, Nov 10 2016

The columns of this triangle are the columns of A277647 with squarefree index.

Other that the first (with length 1), row n has length A278100(n).

			Triangle begins:
1;
2, 1, 1;
3, 2, 1, 1, 1, 1;
4, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1;
...
where the first 11 terms of A005117(k) are
1, 2, 3, 5, 6, 7,10,11,13,14,15.

Jason Kimberley, Table of n, a(n) for n = 1..10716 (the first 37 rows of the triangle)

Cf. A277646, A277647.

A277647:=func;
A277648_row:=funcA277647(n,k):k in[1..n^2]|IsSquarefree(k)]>;
&cat[A277648_row(n):n in[1..8]];

DeleteCases[#, 0] & /@ Table[Boole[SquareFreeQ@ k] Floor[n/Sqrt@ k], {n, 8}, {k, n^2}] (* Michael De Vlieger, Nov 24 2016 *)

row(n)={apply(k->sqrtint(n^2\k), select(issquarefree,[1..n^2]))}
for(n=1, 6, print(row(n))) \\ Andrew Howroyd, Feb 28 2018

T(n,k) = A000196(A277646(n,A005117(k))).

T(n,k) sqrt(A005117(k)) <= n < (T(n,k)+1) sqrt(A005117(k)).

Missing a(3009) in b-file inserted by Andrew Howroyd, Feb 28 2018

A278108 Irregular triangle read by rows: T(n,k) = floor(n/k^2) for 1 <= k^2 <= n.

Original entry on oeis.org

1, 2, 3, 4, 1, 5, 1, 6, 1, 7, 1, 8, 2, 9, 2, 1, 10, 2, 1, 11, 2, 1, 12, 3, 1, 13, 3, 1, 14, 3, 1, 15, 3, 1, 16, 4, 1, 1, 17, 4, 1, 1, 18, 4, 2, 1, 19, 4, 2, 1, 20, 5, 2, 1, 21, 5, 2, 1, 22, 5, 2, 1, 23, 5, 2, 1, 24, 6, 2, 1, 25, 6, 2, 1, 1, 26, 6, 2, 1, 1, 27, 6, 3, 1, 1, 28, 7, 3, 1, 1, 29, 7, 3, 1, 1

Offset: 1

Jason Kimberley, Feb 01 2017

The row length sequence is A000196.

			The first 27 rows are:
1;
2;
3;
4, 1;
5, 1;
6, 1;
7, 1;
8, 2;
9, 2, 1;
10, 2, 1;
11, 2, 1;
12, 3, 1;
13, 3, 1;
14, 3, 1;
15, 3, 1;
16, 4, 1, 1;
17, 4, 1, 1;
18, 4, 2, 1;
19, 4, 2, 1;
20, 5, 2, 1;
21, 5, 2, 1;
22, 5, 2, 1;
23, 5, 2, 1;
24, 6, 2, 1;
25, 6, 2, 1, 1;
26, 6, 2, 1, 1;
27, 6, 3, 1, 1;

Jason Kimberley, Table of n, a(n) for n = 1..20615 (the first 1000 rows of the triangle)

Cf. A000196, A010766, A277646, A277647, A278109.

[n div k^2:k in[1..Isqrt(n)],n in[1..27]];

A278111 Triangle T(n,k) = floor(2n^2/k) for 1 <= k <= 2n^2, read by rows.

Original entry on oeis.org

2, 1, 8, 4, 2, 2, 1, 1, 1, 1, 18, 9, 6, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 32, 16, 10, 8, 6, 5, 4, 4, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 50, 25, 16, 12, 10, 8, 7, 6, 5, 5, 4, 4, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Jason Kimberley, Feb 08 2017

			The first five rows are:
2, 1;
8, 4, 2, 2, 1, 1, 1, 1;
18, 9, 6, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1;
32, 16, 10, 8, 6, 5, 4, 4, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
50, 25, 16, 12, 10, 8, 7, 6, 5, 5, 4, 4, 3, 3, 3, 3, 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;

Jason Kimberley, Table of i, a(i) for i = 1..11050 (T(n,k) for n = 1..25)

Cf. A277646.

A278111:=func;
[A278111(n,k):k in[1..2*n^2],n in[1..5]];

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

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

Formula

Extensions

A277647 Triangle T(n,k) = floor(n/sqrt(k)) for 1 <= k <= n^2, read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A277648 Triangle T(n,k) = A277647(n, A005117(k)), read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A278108 Irregular triangle read by rows: T(n,k) = floor(n/k^2) for 1 <= k^2 <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

A278111 Triangle T(n,k) = floor(2n^2/k) for 1 <= k <= 2n^2, read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Formula