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

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

Author

Ralf Stephan, May 06 2005

Keywords

• nonn

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

• M. F. Hasler, Table of n, a(n) for n = 0..9999

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