A010766 -id:A010766 - cp's OEIS frontend

A134979 Triangle read by rows: T(n,k) = number of partitions of n where the maximum number of objects in partitions of any given size is k.

1, 0, 2, 0, 1, 2, 0, 1, 1, 3, 0, 0, 3, 2, 2, 0, 0, 2, 4, 1, 4, 0, 0, 1, 6, 3, 3, 2, 0, 0, 1, 6, 4, 6, 1, 4, 0, 0, 0, 6, 7, 8, 3, 3, 3, 0, 0, 0, 5, 7, 14, 4, 6, 2, 4, 0, 0, 0, 5, 7, 18, 7, 9, 5, 3, 2, 0, 0, 0, 3, 10, 22, 9, 14, 6, 6, 1, 6, 0, 0, 0, 2, 9, 26, 15, 19, 11, 9, 3, 5, 2

Offset: 1

Franklin T. Adams-Watters, Feb 04 2008

Every column is eventually 0; the last row with a nonzero value in column k is A024916(k). T(A024916(k)-i, k) <= P(i), where P is the partition function (A000041); equality holds for 0 <= i <= k. The partition represented by the last number in column k is row k of A010766.

			For the partition [3,2^2], there are 3 objects in the part of size 3 and 4 objects in the parts of size 2, so this partition is counted towards T(7,4).
Triangle T(n,k) begins:
  1;
  0, 2;
  0, 1, 2;
  0, 1, 1, 3;
  0, 0, 3, 2,  2;
  0, 0, 2, 4,  1,  4;
  0, 0, 1, 6,  3,  3, 2;
  0, 0, 1, 6,  4,  6, 1,  4;
  0, 0, 0, 6,  7,  8, 3,  3, 3;
  0, 0, 0, 5,  7, 14, 4,  6, 2, 4;
  0, 0, 0, 5,  7, 18, 7,  9, 5, 3, 2;
  0, 0, 0, 3, 10, 22, 9, 14, 6, 6, 1, 6;
  ...

Alois P. Heinz, Rows n = 1..200, flattened

Cf. A008284, A091602, A000041 (row sums), A000005 (main diagonal), A032741 (2nd diagonal), A010766.

Column sums give A332233.

b:= proc(n, i, m) option remember; `if`(n=0 or i=1, x^
      max(m, n), add(b(n-i*j, i-1, max(m, i*j)), j=0..n/i))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$2, 0)):
seq(T(n), n=1..20);  # Alois P. Heinz, Feb 07 2020

b[n_, i_, m_] := b[n, i, m] = If[n == 0 || i == 1, x^Max[m, n], Sum[b[n - i j, i - 1, Max[m, i j]], {j, 0, n/i}]];
T[n_] := Table[Coefficient[b[n, n, 0], x, i], {i, 1, n}];
Array[T, 20] // Flatten (* Jean-François Alcover, Nov 10 2020, after Alois P. Heinz *)

A138808 Number of integer pairs (x,y), x > 0, y > 0, such that x <= p, y <= q for any factorization n = p*q.

Original entry on oeis.org

1, 3, 5, 8, 9, 14, 13, 20, 21, 26, 21, 35, 25, 38, 41, 48, 33, 57, 37, 64, 61, 62, 45, 84, 65, 74, 81, 96, 57, 109, 61, 112, 101, 98, 101, 138, 73, 110, 121, 151, 81, 160, 85, 160, 161, 134, 93, 196, 133, 185, 161, 192, 105, 216, 173, 223, 181, 170, 117, 258

Offset: 1

Jonas Wallgren, May 16 2008

nonn

Conjecture: the row sums of the plane partitions A010766 are upper bounds. - R. J. Mathar, Aug 06 2008
a(n) is divisible by n iff n=1 or n belongs to A227993. - Rémy Sigrist, Mar 06 2017
a(n) >= 2*n - 1, with equality iff n is not composite. - Rémy Sigrist, Mar 12 2017

			a(8) = these 20 marked *'s:
-|12345678
-+--------
1|********
2|****
3|**
4|**
5|*
6|*
7|*
8|*

Paul Tek, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Illustration of the first terms

Cf. A227993.

a(n) = my(ar=0, pw=0); fordiv(n, w, ar=ar+(w-pw)*n/w; pw=w); return (ar) \\ Paul Tek, Mar 21 2015

a(n) = n*(m - Sum_{k=1..m-1} d(k)/d(k+1)), where d(1) < d(2) < ... < d(m) denote the divisors of n. - Rémy Sigrist, Mar 06 2017

More terms from Paul Tek, Mar 21 2015

Typo in name corrected by Rémy Sigrist, Mar 05 2017

A278105 a(n) = floor(3/n).

Original entry on oeis.org

3, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Jason Kimberley, Nov 23 2016

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

Cf. A033322, A033324, ..., A033420, A033421, A033422, A033427, A033426, A033425, A033424, A033423.

This sequence is (ignoring the trailing zeros) the third row of A010766.

Magma
```
[3 div n: n in[1..100]];
```

Table[Floor[3/n], {n, 105}] (* Michael De Vlieger, Nov 24 2016 *)

a(n) = A033322(n)+A154272(n). - R. J. Mathar, Jun 21 2025

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).

A348388 Irregular triangle read by rows: T(n, k) = floor((n-k)/k), for k = 1, 2, ..., floor(n/2) and n >= 2.

Original entry on oeis.org

1, 2, 3, 1, 4, 1, 5, 2, 1, 6, 2, 1, 7, 3, 1, 1, 8, 3, 2, 1, 9, 4, 2, 1, 1, 10, 4, 2, 1, 1, 11, 5, 3, 2, 1, 1, 12, 5, 3, 2, 1, 1, 13, 6, 3, 2, 1, 1, 1, 14, 6, 4, 2, 2, 1, 1, 15, 7, 4, 3, 2, 1, 1, 1, 16, 7, 4, 3, 2, 1, 1, 1, 17, 8, 5, 3, 2, 2, 1, 1, 1, 18, 8, 5, 3, 2, 2, 1, 1, 1, 19, 9, 5, 4, 3, 2, 1, 1, 1, 1

Offset: 2

Wolfdieter Lang, Oct 31 2021

This irregular triangle T(n, k) gives the number of multiples of number k, larger than k and not exceeding n, for k = 1, 2, ..., floor(n/2), for n >= 2. See A348389 for the array of these multiples.
The length of row n is floor(n/2) = A004526(n), for n >= 2.
The row sums give A002541(n). See the formula given there by Wesley Ivan Hurt, May 08 2016.
The columns give the k-fold repeated positive integers k, for k >= 1.

			The irregular triangle T(n, k) begins:
n\k   1 2 3 4 5 6 7 8 9 10 ...
------------------------------
2:    1
3:    2
4:    3 1
5:    4 1
6:    5 2 1
7:    6 2 1
8:    7 3 1 1
9:    8 3 2 1
10:   9 4 2 1 1
11:  10 4 2 1 1
12:  11 5 3 2 1 1
13:  12 5 3 2 1 1
14:  13 6 3 2 1 1 1
15:  14 6 4 2 2 1 1
16:  15 7 4 3 2 1 1 1
17:  16 7 4 3 2 1 1 1
18:  17 8 5 3 2 2 1 1 1
19:  18 8 5 3 2 2 1 1 1
20:  19 9 5 4 3 2 1 1 1  1
...

Karl-Heinz Hofmann, Table of n, a(n) for n = 2..10101

Columns k (with varying offsets): A000027, A004526, A008620, A008621, A002266, A097992, ...

Cf. A002541, A004526, A010766, A348389.

T[n_, k_] := Floor[(n - k)/k]; Table[T[n, k], {n, 2, 20}, {k, 1, Floor[n/2]}] // Flatten (* Amiram Eldar, Nov 02 2021 *)

def A348388row(n): return [(n - k) // k for k in range(1, 1 + n // 2)]
for n in range(2, 21): print(A348388row(n))  # Peter Luschny, Nov 05 2021

T(n, k) = floor((n-k)/k), for k = 1, 2, ..., floor(n/2) and n >= 2.

G.f. of column k: G(k, x) = x^(2*k)/((1 - x)*(1 - x^k)).

A004199 Table of [ x/y ], where (x,y) = (1,1),(1,2),(2,1),(1,3),(2,2),(3,1),...

Original entry on oeis.org

1, 0, 2, 0, 1, 3, 0, 0, 1, 4, 0, 0, 1, 2, 5, 0, 0, 0, 1, 2, 6, 0, 0, 0, 1, 1, 3, 7, 0, 0, 0, 0, 1, 2, 3, 8, 0, 0, 0, 0, 1, 1, 2, 4, 9, 0, 0, 0, 0, 0, 1, 1, 2, 4, 10, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 11, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 12, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 3, 6, 13, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 6, 14

Offset: 1

David W. Wilson

Entry in row n and column k is also the number of multiples of k less than or equal to n, n,k >= 1. - L. Edson Jeffery, Aug 31 2014

			Array begins:
  1, 0, 0, 0, 0, 0, 0, 0, ...
  2, 1, 0, 0, 0, 0, 0, 0, ...
  3, 1, 1, 0, 0, 0, 0, 0, ...
  4, 2, 1, 1, 0, 0, 0, 0, ...
  5, 2, 1, 1, 1, 0, 0, 0, ...
  ...

Cf. A002541 (antidiagonal sums).

Cf. A010766 (same sequence as triangle, omitting the zeros), A010783.

(* Array version: *)
Grid[Table[Floor[n/k], {n, 14}, {k, 14}]] (* L. Edson Jeffery, Aug 31 2014 *)
(* Array antidiagonals flattened: *)
Flatten[Table[Floor[(n - k + 1)/k], {n, 14}, {k, n}]] (* L. Edson Jeffery, Aug 31 2014 *)

Sum_{k=1..n} a(n-k+1,k) = A002541(n+1).

A033325 a(n) = floor(5/n).

Original entry on oeis.org

5, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Jeff Burch

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

Cf. A010766.

[Floor(5/n): n in [1..100]]; // Wesley Ivan Hurt, Apr 04 2023

Floor[5/Range[100]] (* Harvey P. Dale, Apr 22 2020 *)

A033326 a(n) = floor(6/n).

Original entry on oeis.org

6, 3, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Jeff Burch

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

Cf. A010766.

[Floor(6/n): n in [1..100]]; // Wesley Ivan Hurt, Apr 04 2023

Floor[6/Range[100]] (* Wesley Ivan Hurt, Mar 19 2023 *)

A033327 a(n) = floor(7/n).

Original entry on oeis.org

7, 3, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Jeff Burch

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

Cf. A010766.

[Floor(7/n): n in [1..100]]; // Wesley Ivan Hurt, Apr 04 2023

Floor[7/Range[100]] (* Wesley Ivan Hurt, Mar 19 2023 *)

A033328 a(n) = floor(8/n).

Original entry on oeis.org

8, 4, 2, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Jeff Burch

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

Cf. A010766.

[Floor(8/n): n in [1..100]]; // Wesley Ivan Hurt, Apr 04 2023

Floor[8/Range[100]] (* Wesley Ivan Hurt, Apr 04 2023 *)

cp's OEIS Frontend

A134979 Triangle read by rows: T(n,k) = number of partitions of n where the maximum number of objects in partitions of any given size is k.

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A138808 Number of integer pairs (x,y), x > 0, y > 0, such that x <= p, y <= q for any factorization n = p*q.

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A278105 a(n) = floor(3/n).

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

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A348388 Irregular triangle read by rows: T(n, k) = floor((n-k)/k), for k = 1, 2, ..., floor(n/2) and n >= 2.

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A004199 Table of [ x/y ], where (x,y) = (1,1),(1,2),(2,1),(1,3),(2,2),(3,1),...

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A033325 a(n) = floor(5/n).

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Magma

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A033326 a(n) = floor(6/n).

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