A033324 -id:A033324 - 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

A046698 a(0) = 0, a(1) = 1, a(n) = a(a(n-1)) + a(a(n-2)) if n > 1.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, R. K. Guy

Partial sums are A004275. Binomial transform is A048492, starting with 0. - Paul Barry, Feb 28 2003
From Elmo R. Oliveira, Jul 25 2024: (Start)
Continued fraction expansion of 2 - sqrt(2) = A101465.
Decimal expansion of 101/9000. (End)

Sequence proposed by Reg Allenby.

Ömür Deveci, Zafer Adıgüzel, and Taha Doğan, On the Generalized Fibonacci-circulant-Hurwitz numbers, Notes on Number Theory and Discrete Mathematics (2020) Vol. 26, No. 1, 179-190.
O. Deveci, Y. Akuzum, E. Karaduman, and O. Erdag, The Cyclic Groups via Bezout Matrices, Journal of Mathematics Research, Vol. 7, No. 2, 2015, pp. 34-41.
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A004275, A048492, A101465 (decimal expansion of 2 - sqrt(2)).

Cf. A033324, A114955, A343461.

CoefficientList[Series[x (1 + x^2)/(1 - x), {x, 0, 104}], x] (* or *)
Nest[Append[#, #[[#[[-1]] + 1]] + #[[#[[-2]] + 1 ]]] &, {0, 1}, 105] (* Michael De Vlieger, Jul 31 2020 *)

PARI
```
a(n)=(n>0)+(n>2)
```

G.f.: x*(1+x^2)/(1-x). - Paul Barry, Feb 28 2003
From Elmo R. Oliveira, Jul 25 2024: (Start)
E.g.f.: 2*exp(x) - x - 1.
a(n) = 2 for n > 2.
a(n) = 2 - A033324(n+2) = 4 - A343461(n+4) = A114955(n+6) - 6. (End)

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

A057075 Table read by antidiagonals of T(n,k)=floor(n^n/k) with n,k >= 1.

Original entry on oeis.org

1, 0, 4, 0, 2, 27, 0, 1, 13, 256, 0, 1, 9, 128, 3125, 0, 0, 6, 85, 1562, 46656, 0, 0, 5, 64, 1041, 23328, 823543, 0, 0, 4, 51, 781, 15552, 411771, 16777216, 0, 0, 3, 42, 625, 11664, 274514, 8388608, 387420489, 0, 0, 3, 36, 520, 9331, 205885, 5592405, 193710244, 10000000000

Offset: 1

Henry Bottomley, Jul 31 2000

			From _Seiichi Manyama_, Aug 12 2023: (Start)
Square array begins:
      1,     0,     0,     0,    0,    0, ...
      4,     2,     1,     1,    0,    0, ...
     27,    13,     9,     6,    5,    4, ...
    256,   128,    85,    64,   51,   42, ...
   3125,  1562,  1041,   781,  625,  520, ...
  46656, 23328, 15552, 11664, 9331, 7776, ... (End)

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Rows are: A000007 (essentially), A033324, A033347, A057066-A057074.
Columns include A000312 and A057065.
Leading diagonal is A000169.
Cf. A060155.

T(n, k) = n^n\k; \\ Seiichi Manyama, Aug 12 2023

A057066 a(n) = floor(4^4/n).

Original entry on oeis.org

256, 128, 85, 64, 51, 42, 36, 32, 28, 25, 23, 21, 19, 18, 17, 16, 15, 14, 13, 12, 12, 11, 11, 10, 10, 9, 9, 9, 8, 8, 8, 8, 7, 7, 7, 7, 6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2

Offset: 1

Henry Bottomley, Jul 31 2000

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

Cf. A000007, A033324, A033347, A057067-A057075.

a(n) = 4^4\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).

A057067 a(n) = floor(5^5/n).

Original entry on oeis.org

3125, 1562, 1041, 781, 625, 520, 446, 390, 347, 312, 284, 260, 240, 223, 208, 195, 183, 173, 164, 156, 148, 142, 135, 130, 125, 120, 115, 111, 107, 104, 100, 97, 94, 91, 89, 86, 84, 82, 80, 78, 76, 74, 72, 71, 69, 67, 66, 65, 63, 62, 61, 60, 58, 57, 56, 55, 54

Offset: 1

Henry Bottomley, Jul 31 2000

nonn

a(n) = 1 for 1563 <= n <= 3125, and is 0 for values of n >= 5^5 = 3126.

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

Cf. A000007, A033324, A033347, A057066-A057075.

A057067:=n->floor(5^5/n); seq(A057067(n), n=1..100); # Wesley Ivan Hurt, Nov 27 2013

Table[Floor[5^5/n], {n,100}] (* Wesley Ivan Hurt, Nov 27 2013 *)

n->5^5\n  \\ - M. F. Hasler, Nov 30 2013

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

A057074 a(n) = floor(12^12/n).

Original entry on oeis.org

8916100448256, 4458050224128, 2972033482752, 2229025112064, 1783220089651, 1486016741376, 1273728635465, 1114512556032, 990677827584, 891610044825, 810554586205, 743008370688, 685853880635, 636864317732, 594406696550

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. A000007, A033324, A033347, A057066-A057075.

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

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

A057068 a(n) = floor(6^6/n).

Original entry on oeis.org

46656, 23328, 15552, 11664, 9331, 7776, 6665, 5832, 5184, 4665, 4241, 3888, 3588, 3332, 3110, 2916, 2744, 2592, 2455, 2332, 2221, 2120, 2028, 1944, 1866, 1794, 1728, 1666, 1608, 1555, 1505, 1458, 1413, 1372, 1333, 1296, 1260, 1227, 1196, 1166, 1137

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. A000007, A033324, A033347, A057066-A057075.

a(n) = 6^6\n; \\ Seiichi Manyama, Aug 12 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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A046698 a(0) = 0, a(1) = 1, a(n) = a(a(n-1)) + a(a(n-2)) if n > 1.

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

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A057075 Table read by antidiagonals of T(n,k)=floor(n^n/k) with n,k >= 1.

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A057066 a(n) = floor(4^4/n).

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

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Magma

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A057067 a(n) = floor(5^5/n).

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A057074 a(n) = floor(12^12/n).

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