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 1-5 of 5 results.

A118013 Triangle read by rows: T(n,k) = floor(n^2/k), 1<=k<=n.

Original entry on oeis.org

1, 4, 2, 9, 4, 3, 16, 8, 5, 4, 25, 12, 8, 6, 5, 36, 18, 12, 9, 7, 6, 49, 24, 16, 12, 9, 8, 7, 64, 32, 21, 16, 12, 10, 9, 8, 81, 40, 27, 20, 16, 13, 11, 10, 9, 100, 50, 33, 25, 20, 16, 14, 12, 11, 10, 121, 60, 40, 30, 24, 20, 17, 15, 13, 12, 11, 144, 72, 48, 36, 28, 24, 20, 18, 16, 14
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 10 2006

Keywords

Comments

T(n,1) = A000290(n); T(n,n) = n;
T(n,2) = A007590(n) for n>1;
T(n,3) = A000212(n) for n>2;
T(n,4) = A002620(n) for n>3;
T(n,5) = A118015(n) for n>4;
T(n,6) = A056827(n) for n>5;
central terms give A008574: T(2*k-1,k) = 4*(k-1)+0^(k-1);
row sums give A118014.

Examples

			Triangle begins:
1,
4, 2,
9, 4, 3,
16, 8, 5, 4,

Links

Crossrefs

Cf. A010766.

Programs

  • Haskell
    a118013 n k = a118013_tabl !! (n-1) !! (k-1)
a118013_row n = map (div (n^2)) [1..n]
a118013_tabl = map a118013_row [1..]
-- Reinhard Zumkeller, Jan 22 2012
  • PARI
    T(n,k)=n^2\k \\ Charles R Greathouse IV, Jan 15 2012

A153817 a(n)=Sum_{k=1..n} floor((n*k)/(n+k)).

Original entry on oeis.org

0, 1, 2, 4, 6, 10, 13, 18, 22, 28, 33, 42, 48, 57, 65, 74, 83, 95, 105, 118, 130, 142, 155, 171, 185, 200, 216, 232, 248, 270, 286, 307, 325, 345, 365, 390, 409, 433, 454, 480, 502, 532, 555, 583, 610, 636, 665, 695, 723, 753, 784, 814, 847, 880, 913, 947, 981
Offset: 1

Views

Author

Ctibor O. Zizka, Jan 02 2009

Keywords

Crossrefs

Cf. A000005, A006218, A118014, A153818.

Programs

  • Maple
    A153817 := proc(n) add(floor(n*k/(n+k)), k=1..n) ; end: seq(A153817(n),n=1..100) ; # R. J. Mathar, Feb 21 2009
  • Mathematica
    Table[Sum[Floor[n*k/(n + k)], {k, 1, n}], {n, 1, 100}] (* Vaclav Kotesovec, May 28 2021 *)

Formula

a(n) ~ (1 - log(2)) * n^2. - Vaclav Kotesovec, May 28 2021

Extensions

More terms from R. J. Mathar, Feb 21 2009

A156745 a(n) = Sum_{k=1..n} floor((n+k)/k) = n + Sum_{k=1..n} sigma_0(k), where sigma_0(k) is A000005(k). Also a(n) = n + A006218(n).

Original entry on oeis.org

2, 5, 8, 12, 15, 20, 23, 28, 32, 37, 40, 47, 50, 55, 60, 66, 69, 76, 79, 86, 91, 96, 99, 108, 112, 117, 122, 129, 132, 141, 144, 151, 156, 161, 166, 176, 179, 184, 189, 198, 201, 210, 213, 220, 227, 232, 235, 246, 250, 257, 262, 269, 272, 281, 286, 295, 300
Offset: 1

Views

Author

Ctibor O. Zizka, Feb 14 2009

Keywords

Comments

Generalized sequence b(n) = Sum_{k=1..n} floor((n+k*t)/k) = t*n + Sum_{k=1..n} sigma_0(k), where sigma_0(k) is A000005(k). Also b(n) = t*n + A006218(n).
Partial sums of A334954. - Omar E. Pol, Sep 26 2020

Crossrefs

Cf. A000005, A006218, A153818, A118014, A002541, A334954.

Programs

  • PARI
    a(n) = n + sum(k=1, n, numdiv(k)); \\ Michel Marcus, Oct 02 2020
  • Python
    from math import isqrt
def A156745(n): return n-(s:=isqrt(n))**2+(sum(n//k for k in range(1,s+1))<<1) # Chai Wah Wu, Oct 23 2023

Formula

a(n) = 2*n + Sum_{k=1..floor(n/2)} floor((n-k)/k). - Wesley Ivan Hurt, Dec 25 2020
a(n) = A005843(n) + A002541(n), after Wesley Ivan Hurt. - Omar E. Pol, Dec 25 2020

Extensions

More terms from Eric M. Schmidt, Feb 28 2014

A069627 Sum_{k=1..n} floor(n*(n-1)/(2*k)).

Original entry on oeis.org

0, 0, 1, 5, 12, 22, 35, 53, 74, 101, 129, 162, 202, 244, 292, 344, 403, 463, 527, 601, 676, 762, 844, 937, 1035, 1138, 1245, 1355, 1476, 1597, 1726, 1862, 2002, 2149, 2300, 2454, 2621, 2784, 2957, 3136, 3323, 3515, 3707, 3914, 4119, 4338, 4551, 4782, 5012, 5250
Offset: 0

Views

Author

N. J. A. Sloane, Oct 28 2008

Keywords

Comments

The old entry with this sequence number was a duplicate of A072137.

Crossrefs

Cf. A006218, A118014.

Programs

  • Maple
    f:=n->add(floor( n*(n-1)/(2*k) ),k=1..n );

Formula

a(n) ~ n^2 * (gamma + log(n))/2, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Dec 23 2020

A211783 Rectangular array: R(n,k)=n^2+[(n^2)/2]+...+[(n^2)/k], where [ ]=floor, by antidiagonals.

Original entry on oeis.org

1, 4, 1, 9, 6, 1, 16, 13, 7, 1, 25, 24, 16, 8, 1, 36, 37, 29, 18, 8, 1, 49, 54, 45, 33, 19, 8, 1, 64, 73, 66, 51, 36, 20, 8, 1, 81, 96, 89, 75, 56, 38, 21, 8, 1, 100, 121, 117, 101, 82, 60, 40, 22, 8, 1, 121, 150, 148, 133, 110, 88, 63, 42, 23, 8, 1, 144, 181, 183
Offset: 0

Views

Author

Clark Kimberling, Apr 20 2012

Keywords

Comments

For n>=1, row n is a homogeneous linear recurrence sequence with palindromic recurrence coefficients in the sense described at A211701.
Row 1: A000290
Row 2: A032528
Row 3: A211784
R(n,n)=A118014(n,n)
The sequence approached as a limit of the rows is A175346: (1,8,23,50,87,140,...)

Examples

			Northwest corner:
1....4....9....16....25....36
1....6....13...24....37....54
1....7....16...29....35....66
1....8....18...33....51....75
1....8....19...36....56....82
1....8....20...38....60....88
1....8....21...40....63....93

Crossrefs

Cf. A211701

Programs

  • Mathematica
    f[n_, m_] := Sum[Floor[n^2/k], {k, 1, m}]
TableForm[Table[f[n, m], {m, 1, 40}, {n, 1, 16}]]
Flatten[Table[f[n + 1 - m, m], {n, 1, 14}, {m, 1, n}]]
Showing 1-5 of 5 results.

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

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