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 1-8 of 8 results.

A078153 a(n) = A051201(n) - A000203(n).

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 5, 0, 6, 3, 10, 0, 15, 7, 9, 8, 22, 4, 24, 9, 21, 19, 32, 0, 35, 26, 30, 17, 44, 11, 52, 24, 41, 37, 45, 12, 66, 46, 52, 22, 71, 27, 80, 43, 52, 60, 85, 14, 89, 56, 79, 56, 101, 39, 89, 52, 94, 86, 117, 15, 122, 90, 85, 73, 118, 62, 139, 84, 116, 72, 145, 36
Offset: 1

Views

Author

Labos Elemer, Nov 27 2002

Keywords

Examples

			n=15: sequence of D1 = {floor(15/j)} = {15,7,5,3,3,2,2,1,1,1,1,1,1,1,1}, Union(D1) = {15,7,5,3,2,1} = divisors(15) and {7,2}, a(15) = (15+7+5+3+2+1) - sigma(15) = 7 + 2 = 9.

Links

Crossrefs

Cf. A051201, A000203, A055086, A000005, A078152, A076891.

Programs

  • Mathematica
    Table[Apply[Plus, Union[Table[Floor[w/j], {j, 1, w}]]] -DivisorSigma[1, w], {w, 1, 128}]
  • PARI
    a(n)=my(m=(sqrtint(4*n+1)-1)\2); m*(m+1)/2+sum(k=1, n\(m+1), n\k)-sigma(n) \\ Charles R Greathouse IV, Feb 14 2013

A078162 a(n) = A051201(n!).

Original entry on oeis.org

1, 1, 3, 12, 60, 406, 3120, 26823, 257022, 2712926, 31311886, 392308000, 5302926524, 76924127104, 1191973363059, 19650226851964, 343408795841627, 6341818750193544, 123405357866753108, 2523790088462591703, 54119963305579115810, 1214292989338825766682
Offset: 0

Views

Author

Labos Elemer, Nov 27 2002

Keywords

Crossrefs

Cf. A051201, A078159, A078160, A078161.

Programs

  • Mathematica
    Table[ Apply[ Plus, Union[ Table[ Floor[w!/j], {j, 1, w!}]]], {w, 1, 11}]

Extensions

Extended by Robert G. Wilson v, Dec 02 2002
Terms a(12) onward from Max Alekseyev, Feb 12 2012
a(0)=1 prepended by Alois P. Heinz, Oct 31 2023

A078161 a(n) = A051201(2^n).

Original entry on oeis.org

1, 3, 7, 15, 39, 87, 200, 438, 981, 2135, 4639, 9971, 21424, 45699, 97096, 205563, 433895, 913244, 1917491, 4016704, 8397235, 17521118, 36497171, 75900377, 157619705, 326868209, 676998896, 1400510860, 2894068715, 5974185368, 12320552102, 25385332979
Offset: 0

Views

Author

Labos Elemer, Nov 27 2002

Keywords

Crossrefs

Cf. A051201, A078159, A078160, A078162.

Programs

  • Mathematica
    Table[ Apply[ Plus, Union[ Table[ Floor[2^w/j], {j, 1, 2^w}]]], {w, 1, 25}]

Extensions

Extended by Robert G. Wilson v, Dec 02 2002
Terms a(26) onward from Max Alekseyev, Feb 12 2012
a(0)=1 prepended by Alois P. Heinz, Oct 31 2023
a(21) corrected by Sean A. Irvine, Jun 18 2025

A078172 a(n) = A051201(A000040(n)).

Original entry on oeis.org

3, 4, 8, 13, 22, 29, 40, 44, 56, 74, 84, 104, 113, 124, 133, 155, 177, 184, 207, 217, 232, 247, 266, 283, 316, 337, 342, 353, 360, 382, 434, 445, 475, 480, 522, 529, 561, 579, 589, 622, 639, 649, 692, 700, 725, 732, 787, 823, 851, 858, 871, 889, 915, 944, 982
Offset: 1

Views

Author

Labos Elemer, Nov 27 2002

Keywords

Examples

			n=25:p(25)=97, sequence Union[Floor[97/j]]= {1,2,3,4,5,6,7,8,9,10,12,13,16,19,24,32,48,97}, sum=a[25]=316.

Crossrefs

Cf. A000040, A055086, A051201, A078171.

Programs

  • Mathematica
    Table[Apply[Plus, Union[Table[Floor[Prime[w]/j], {j, 1, Prime[w]}]]], {w, 1, 100}]

A078163 a(n) = A051201(n^2).

Original entry on oeis.org

1, 7, 19, 39, 66, 103, 146, 200, 263, 336, 418, 511, 611, 724, 848, 981, 1125, 1281, 1446, 1624, 1811, 2010, 2220, 2446, 2676, 2922, 3178, 3450, 3730, 4023, 4324, 4639, 4967, 5310, 5665, 6029, 6404, 6794, 7199, 7614, 8041, 8479, 8930, 9394, 9874
Offset: 1

Views

Author

Labos Elemer, Nov 27 2002

Keywords

Crossrefs

Cf. A000290, A051201, A078161, A078162.

Programs

  • Mathematica
    Table[Apply[Plus, Union[Table[Floor[w^2/j], {j, 1, w^2}]]], {w, 1, 50}]

A078152 a(n) = A055086(n) - A000005(n).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 2, 0, 2, 1, 3, 0, 4, 2, 2, 2, 5, 1, 5, 2, 4, 4, 6, 0, 6, 5, 5, 3, 7, 2, 8, 4, 6, 6, 6, 2, 9, 7, 7, 3, 9, 4, 10, 6, 6, 8, 10, 2, 10, 7, 9, 7, 11, 5, 9, 6, 10, 10, 12, 2, 12, 10, 8, 8, 11, 7, 13, 9, 11, 7, 13, 4, 14, 12, 10, 10, 12, 8, 14, 6, 12, 13, 15, 5, 13, 13, 13, 9, 15, 6, 14
Offset: 1

Views

Author

Labos Elemer, Nov 27 2002

Keywords

Comments

It appears that the indices of the zeros in the sequence are in A018253. - Omar E. Pol, Oct 22 2013

Examples

			n=15: sequence of D1 = {floor(15/j)} = {15,7,5,3,3,2,2,1,1,1,1,1,1,1,1}, Union(D1) = {15,7,5,3,2,1} = divisors(15) and {7,2}, a(15)=2 the number of terms beyond divisors.

Crossrefs

Cf. A055086, A000005, A051201, A000203, A076891.

Programs

  • Mathematica
    Table[Length[Union[Table[Floor[w/j], {j, 1, w}]]] -DivisorSigma[0, w], {w, 1, 128}]

A078171 a(n) = A055086(A000040(n)).

Original entry on oeis.org

2, 2, 3, 4, 5, 6, 7, 7, 8, 9, 10, 11, 11, 12, 12, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 19, 19, 19, 19, 20, 21, 21, 22, 22, 23, 23, 24, 24, 24, 25, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 29, 29, 30, 30, 31, 31, 31, 31, 32, 32, 32, 33, 34, 34, 34, 34, 35, 35, 36, 36, 36, 36
Offset: 1

Views

Author

Labos Elemer, Nov 27 2002

Keywords

Examples

			n=25:p(25)=97, sequence Union[Floor[97/j]]= {1,2,3,4,5,6,7,8,9,10,12,13,16,19,24,32,48,97}, a[25]=18.

Crossrefs

Cf. A000040, A055086, A051201, A078172.

Programs

  • Mathematica
    Table[Length[Union[Table[Floor[Prime[w]/j], {j, 1, Prime[w]}]]], {w, 1, 100}]

A210256 Differences of the sum of distinct values of {floor(n/k), k=1,...,n}.

Original entry on oeis.org

2, 1, 3, 1, 4, 1, 2, 4, 2, 1, 6, 1, 2, 2, 6, 1, 3, 1, 7, 2, 2, 1, 4, 6, 2, 2, 3, 1, 9, 1, 3, 2, 2, 2, 10, 1, 2, 2, 4, 1, 10, 1, 3, 3, 2, 1, 5, 8, 3, 2, 3, 1, 4, 2, 11, 2, 2, 1, 6, 1, 2, 3, 11, 2, 4, 1, 3, 2, 4, 1, 14, 1, 2, 3, 3, 2, 4, 1, 5, 11, 2, 1, 6, 2, 2
Offset: 1

Views

Author

John W. Layman, Mar 19 2012

Keywords

Comments

Differences of A051201.
It appears that a(n)=1 if and only if n>1 and n+1 is a prime. For example, the indices where 1 occurs in {a(n)} are {2,4,6,10,12,16,...}. Adding 1 to each of these gives {3,5,7,11,13,17,...} each of which is a prime.

Links

Crossrefs

Cf. A006218, A051201, A055086.

Programs

  • Maple
    b:= proc(n) option remember; add(i, i={seq(floor(n/k), k=1..n)}) end:
a:= n-> b(n+1)-b(n):
seq(a(n), n=1..150); # Alois P. Heinz, Mar 19 2012
  • Mathematica
    b[n_] := b[n] = Total@ Union@ Table[Floor[n/k], {k, 1, n}];
a[n_] := b[n+1] - b[n];
Array[a, 150] (* Jean-François Alcover, Nov 20 2020, after Alois P. Heinz *)
  • Python
    from math import isqrt
def A210256(n): return ((m:=isqrt((n+1<<2)+1)+1>>1)*(m-1)>>1)+sum((n+1)//k for k in range(1,(n+1)//m+1))-((r:=isqrt((n<<2)+1)+1>>1)*(r-1)>>1)-sum(n//k for k in range(1,n//r+1)) # Chai Wah Wu, Oct 31 2023
Showing 1-8 of 8 results.

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

Content is available under The OEIS End-User License Agreement.