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.

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A106847 a(n) = Sum {k + j*m <= n} (k + j*m), with 0 < k,j,m <= n.

Original entry on oeis.org

0, 0, 2, 11, 31, 71, 131, 229, 357, 537, 767, 1064, 1412, 1867, 2385, 3000, 3720, 4570, 5506, 6608, 7808, 9194, 10734, 12436, 14260, 16360, 18622, 21079, 23739, 26668, 29758, 33199, 36815, 40742, 44924, 49369, 54085, 59265, 64661, 70355
Offset: 0

Views

Author

Ralf Stephan, May 06 2005

Keywords

Examples

			We have 1+1*1=2<=3, 1+2*1=3, 1+1*2=3, 2+1*1=3, thus a(3)=2+3+3+3=11.

Links

Crossrefs

Cf. A106633, A106634, A106846, A078567 (number of terms).
Cf. A006218, A143272, A143274, A143127, A319085.

Programs

  • Maple
    A106847 := proc(n)
    local a,k,l,m ;
    a := 0 ;
    for k from 1 to n do
        for l from 1 to n-k do
            m := floor((n-k)/l) ;
            if m >=1 then
                m := min(m,n) ;
                a := a+m*k+l*m*(m+1)/2 ;
            end if;
        end do:
    end do:
    a ;
end proc: # R. J. Mathar, Oct 17 2012
  • Mathematica
    A106847[n_] := Module[{a, k, l, m}, a = 0; For[k = 1, k <= n, k++, For[l = 1, l <= n - k, l++, If[l == 0, m = n, m = Floor[(n - k)/l]]; If[m >= 1, m = Min[m, n]; a = a + m*k + l*m*(m + 1)/2]]]; a];
Table[A106847[n], {n, 0, 40}] (* Jean-François Alcover, Apr 04 2024, after R. J. Mathar *)
  • PARI
    A106847(n)=sum(m=1,n-1,sum(k=1,(n-1)\m,(n-m*k)*(n+m*k+1)))/2  \\ M. F. Hasler, Oct 17 2012

Formula

From Ridouane Oudra, Jun 02 2024: (Start)
a(n) = (1/2)*Sum_{k=1..n} (n^2 + n - k^2 - k)*tau(k);
a(n) = (1/2)*(n^2 + n)*A006218(n) - Sum_{k=1..n} A143272(k);
a(n) = (1/2)*((n + 1)*A143274(n) - A143127(n) - A319085(n)). (End)
a(n) ~ n^3 * (log(n) + 2*gamma - 4/3)/3, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jun 15 2024

A143271 Triangle read by rows: A130209 * A000012 * A127648.

Original entry on oeis.org

1, 2, 4, 2, 4, 6, 3, 6, 9, 12, 2, 4, 6, 8, 10, 4, 8, 12, 16, 20, 24, 2, 4, 6, 8, 10, 12, 14, 4, 8, 12, 16, 20, 24, 28, 32, 3, 6, 9, 12, 15, 18, 21, 24, 27, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72
Offset: 1

Views

Author

Gary W. Adamson, Aug 03 2008

Keywords

Comments

Row sums = A143272: (1, 6, 12, 30, 30, 84, 56, ...).
Left border = A000005: (1, 2, 2, 3, 2, 4, 2, 4, 3, ...).

Examples

			First few rows of the triangle =
  1;
  2, 4;
  2, 4,  6;
  3, 6,  9, 12;
  2, 4,  6,  8, 10;
  4, 8, 12, 16, 20, 24;
  2, 4,  6,  8, 10, 12, 14;
  ...
T(5,3) = 6 = 2*3 = d(5)*3.

Crossrefs

Cf. A000005, A127648, A130209, A143272.

Programs

  • PARI
    tabl(nn) = for (n=1, nn, for (k=1, n, print1(numdiv(n)*k, ", "))); \\ Michel Marcus, Jun 05 2023

Formula

T(n,k) = d(n)*k.

Extensions

a(62) corrected by Georg Fischer, Jun 05 2023
Showing 1-2 of 2 results.

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

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