cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

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

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

A375924 Number A(n,k) of partitions of [n] such that the element sum of each block is one more than a multiple of k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 1, 5, 0, 1, 1, 0, 2, 15, 0, 1, 1, 0, 0, 4, 52, 0, 1, 1, 0, 1, 1, 10, 203, 0, 1, 1, 0, 1, 2, 3, 28, 877, 0, 1, 1, 0, 0, 0, 3, 9, 96, 4140, 0, 1, 1, 0, 0, 0, 0, 1, 17, 320, 21147, 0, 1, 1, 0, 0, 0, 1, 1, 8, 108, 1436, 115975, 0
Offset: 0

Views

Author

Alois P. Heinz, Sep 02 2024

Keywords

Examples

			A(5,2) = 10: 12345, 124|3|5, 12|34|5, 12|3|45, 14|23|5, 1|234|5, 1|23|45, 14|25|3, 1|245|3, 1|25|34.
A(6,3) = 9: 136|25|4, 13|256|4, 13|25|46, 16|235|4, 1|2356|4, 1|235|46, 16|25|34, 1|256|34, 1|25|346.
A(7,4) = 8: 14|23|5|67, 1|234|5|67, 1|23|45|67, 1|23|467|5, 14|27|36|5, 1|247|36|5, 1|27|346|5, 1|27|36|45.
A(8,5) = 1: 12345678.
A(8,8) = 4: 18|27|36|45, 1|278|36|45, 1|27|368|45, 1|27|36|458.
A(9,6) = 87: 123469|58|7, 12349|568|7, 12349|58|67, 123568|49|7, ..., 1|25|346789, 16|289|3457, 1|2689|3457, 1|289|34567.
A(9,8) = 5: 18|27|36|45|9, 1|278|36|45|9, 1|27|368|45|9, 1|27|36|458|9, 1|27|36|45|89.
A(9,10) = 1: 1|29|38|47|56.
Square array A(n,k) begins:
  1,      1,    1,    1,   1,  1,  1,  1, 1, 1, 1, ...
  1,      1,    1,    1,   1,  1,  1,  1, 1, 1, 1, ...
  0,      2,    1,    0,   0,  0,  0,  0, 0, 0, 0, ...
  0,      5,    2,    0,   1,  1,  0,  0, 0, 0, 0, ...
  0,     15,    4,    1,   2,  0,  0,  0, 1, 1, 0, ...
  0,     52,   10,    3,   3,  0,  1,  1, 0, 0, 0, ...
  0,    203,   28,    9,   1,  1,  3,  0, 0, 1, 1, ...
  0,    877,   96,   17,   8, 15,  4,  0, 1, 1, 0, ...
  0,   4140,  320,  108,  32,  1,  0,  1, 4, 0, 0, ...
  0,  21147, 1436,  324,  51, 10, 87, 72, 5, 0, 1, ...
  0, 115975, 5556, 1409, 621, 50,  1,  0, 0, 1, 5, ...

Links

Crossrefs

Columns k=0-10 give: A019590(n+1), A000110, A369079, A374692, A375939, A375940, A375941, A375942, A375943, A375944, A375957.
Rows n=1-2 give: A000012, A033322 (for k>=1).
Main diagonal gives A142150 (for n>=2).
A(n+1,n) gives A158416 (for n>=2).
A(n,n+1) gives A135528(n+1).

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

Views

Author

Jason Kimberley, Nov 23 2016

Keywords

Links

Crossrefs

Cf. A033322, A033324, ..., A033420, A033421, A033422, A033427, A033426, A033425, A033424, A033423.
This sequence is (ignoring the trailing zeros) the third row of A010766.

Programs

  • Magma
    [3 div n: n in[1..100]];
  • Mathematica
    Table[Floor[3/n], {n, 105}] (* Michael De Vlieger, Nov 24 2016 *)

Formula

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

A347769 a(0) = 0; a(1) = 1; for n > 1, a(n) = A001065(a(n-1)) = sigma(a(n-1)) - a(n-1) (the sum of aliquot parts of a(n-1)) if this is not yet in the sequence; otherwise a(n) is the smallest number missing from the sequence.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 16, 15, 13, 14, 17, 18, 21, 19, 20, 22, 23, 24, 36, 55, 25, 26, 27, 28, 29, 30, 42, 54, 66, 78, 90, 144, 259, 45, 33, 31, 32, 34, 35, 37, 38, 39, 40, 50, 43, 41, 44, 46, 47, 48, 76, 64, 63, 49, 51, 52, 53, 56, 57, 58, 59, 60, 108, 172
Offset: 0

Views

Author

Eric Chen, Sep 13 2021

Keywords

Comments

This sequence is a permutation of the nonnegative integers iff Catalan's aliquot sequence conjecture (also called Catalan-Dickson conjecture) is true.
a(563) = 276 is the smallest number whose aliquot sequence has not yet been fully determined.
As long as the aliquot sequence of 276 is not known to be finite or eventually periodic, a(563+k) = A008892(k).

Examples

			a(0) = 0, a(1) = 1;
since A001065(a(1)) = 0 has already appeared in this sequence, a(2) = 2;
since A001065(a(2)) = 1 has already appeared in this sequence, a(3) = 3;
...
a(11) = 11;
since A001065(a(11)) = 1 has already appeared in this sequence, a(12) = 12;
since A001065(a(12)) = 16 has not yet appeared in this sequence, a(13) = A001065(a(12)) = 16;
since A001065(a(13)) = 15 has not yet appeared in this sequence, a(14) = A001065(a(13)) = 15;
since A001065(a(14)) = 9 has already appeared in this sequence, a(15) = 13;
...

Links

Crossrefs

Cf. A032451.
Cf. A001065 (sum of aliquot parts).
Cf. A003023, A044050, A098007, A098008: ("length" of aliquot sequences, four versions).
Cf. A007906.
Cf. A115060 (maximum term of aliquot sequences).
Cf. A115350 (termination of the aliquot sequences).
Cf. A098009, A098010 (records of "length" of aliquot sequences).
Cf. A290141, A290142 (records of maximum term of aliquot sequences).
Cf. A000396, A063990, A122726, A206708, A347770.
Cf. A080907, A063769, A121507, A121508, A131884, A126016.
Aliquot sequences starting at various numbers: A000004 (0), A000007 (1), A033322 (2), A010722 (6), A143090 (12), A143645 (24), A010867 (28), A008885 (30), A143721 (38), A008886 (42), A143722 (48), A143723 (52), A008887 (60), A143733 (62), A143737 (68), A143741 (72), A143754 (75), A143755 (80), A143756 (81), A143757 (82), A143758 (84), A143759 (86), A143767 (87), A143846 (88), A143847 (96), A143919 (100), A008888 (138), A008889 (150), A008890 (168), A008891 (180), A203777 (220), A008892 (276), A014360 (552), A014361 (564), A074907 (570), A014362 (660), A269542 (702), A045477 (840), A014363 (966), A014364 (1074), A014365 (1134), A074906 (1521), A143930 (3630), A072891 (12496), A072890 (14316), A171103 (46758), A072892 (1264460).

Programs

  • PARI
    A347769_list(N)=print1(0, ", "); if(N>0, print1(1, ", ")); v=[0, 1]; b=1; for(n=2, N, if(setsearch(Set(v), sigma(b)-b), k=1; while(k<=n, if(!setsearch(Set(v), k), b=k; k=n+1, k++)), b=sigma(b)-b); print1(b, ", "); v=concat(v, b))

A372873 Triangle read by rows: T(n,k) is the number of flattened Catalan words of length n with exactly k runs of descents.

Original entry on oeis.org

1, 0, 2, 0, 1, 4, 0, 0, 6, 8, 0, 0, 1, 24, 16, 0, 0, 0, 10, 80, 32, 0, 0, 0, 1, 60, 240, 64, 0, 0, 0, 0, 14, 280, 672, 128, 0, 0, 0, 0, 1, 112, 1120, 1792, 256, 0, 0, 0, 0, 0, 18, 672, 4032, 4608, 512, 0, 0, 0, 0, 0, 1, 180, 3360, 13440, 11520, 1024
Offset: 1

Views

Author

Stefano Spezia, May 15 2024

Keywords

Examples

			The triangle begins:
  1;
  0, 2;
  0, 1, 4;
  0, 0, 6,  8;
  0, 0, 1, 24, 16;
  0, 0, 0, 10, 80,  32;
  0, 0, 0,  1, 60, 240,   64;
  0, 0, 0,  0, 14, 280,  672,  128;
  0, 0, 0,  0,  1, 112, 1120, 1792, 256;
  ...
T(4,3) = 6 since there 6 flattened Catalan words of length 4 with 3 runs of descents: 0010, 0100, 0101, 0110, 0120, and 0121.

Links

  • Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, Flattened Catalan Words, arXiv:2405.05357 [math.CO], 2024. See p. 11.

Crossrefs

Cf. A000007, A000079, A001788, A007051 (row sums), A033322, A372874.

Programs

  • Mathematica
    T[n_,k_]:=SeriesCoefficient[x*y*(1-2*x*y)/(1-4*x*y-x^2*y+4x^2*y^2),{x,0,n},{y,0,k}]; Table[T[n,k],{n,11},{k,n}]//Flatten (* or *)
T[n_,k_]:=2^(2*k-n-1)*Binomial[n-1, 2*(n-k)]; Table[T[n,k],{n,11},{k,n}]//Flatten

Formula

G.f.: x*y*(1 - 2*x*y)/(1 - 4*x*y - x^2*y + 4*x^2*y^2).
T(n,k) = 2^(2*k-n-1)*binomial(n-1, 2*(n-k)).
T(n,n) = A000079(n-1).
T(n,n-1) = A001788(n-2).
T(n,1) = A000007(n-1).
T(n,2) = A033322(n-1).
Sum_{k>=0} T(n,k) = A007051(n-1).

A303810 Mirror image of the triangle A026794.

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 0, 1, 3, 1, 0, 0, 1, 5, 1, 0, 0, 1, 2, 7, 1, 0, 0, 0, 1, 2, 11, 1, 0, 0, 0, 1, 1, 4, 15, 1, 0, 0, 0, 0, 1, 2, 4, 22, 1, 0, 0, 0, 0, 1, 1, 2, 7, 30, 1, 0, 0, 0, 0, 0, 1, 1, 3, 8, 42, 1, 0, 0, 0, 0, 0, 1, 1, 2, 4, 12, 56, 1, 0, 0, 0, 0, 0, 0, 1, 1, 2, 5, 14, 77, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 3
Offset: 1

Views

Author

Omar E. Pol, May 05 2018

Keywords

Examples

			Triangle begins:
1;
1,  1;
1,  0,  2;
1,  0,  1,  3;
1,  0,  0,  1,  5;
1,  0,  0,  1,  2,  7;
1,  0,  0,  0,  1,  2, 11;
1,  0,  0,  0,  1,  1,  4, 15;
1,  0,  0,  0,  0,  1,  2,  4, 22;
1,  0,  0,  0,  0,  1,  1,  2,  7, 30;
1,  0,  0,  0,  0,  0,  1,  1,  3,  8, 42;
1,  0,  0,  0,  0,  0,  1,  1,  2,  4, 12, 56;
1,  0,  0,  0,  0,  0,  0,  1,  1,  2,  5, 14, 77;
1,  0,  0,  0,  0,  0,  0,  1,  1,  1,  3,  6, 21, 101;
1,  0,  0,  0,  0,  0,  0,  0,  1,  1,  2,  3,  9,  24, 135;
...

Crossrefs

Leading diagonal gives A000041.
Second diagonal gives A002865.
Row sums give A000041, n >= 1.
Columns 1-4: A000012, A000007, A033322, A278105.
Cf. A026794.
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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