A106846 a(n) = Sum_{k + l*m <= n} (k + l*m), with 0 <= k,l,m <= n.

0, 4, 22, 64, 144, 269, 461, 720, 1072, 1522, 2092, 2774, 3626, 4614, 5776, 7126, 8694, 10445, 12461, 14684, 17204, 19997, 23077, 26412, 30156, 34206, 38600, 43352, 48532, 54042, 60072, 66458, 73338, 80664, 88450, 96710, 105638, 114999

Offset

0,2

Links

R. J. Mathar, Table of n, a(n) for n = 0..1000

Formula

From Ridouane Oudra, Jun 24 2024: (Start)

a(n) = (1/2) * (n*(n+1)*(2*n+1) + Sum_{k=1..n} (n^2 + n + k - k^2) * tau(k)).

a(n) = (1/2) * (A055112(n) + (n^2 + n) * A006218(n) + A143127(n) - A319085(n)).

a(n) = A059270(n) + A143127(n) + A106847(n). (End)

Maple

A106846 := proc(n)
    local a, k, l, m ;
    a := 0 ;
    for k from 0 to n do
        for l from 0 to n do
            if l = 0 then
                m := n ;
            else
                m := floor((n-k)/l) ;
            end if;
            if m >=0 then
                m := min(m, n) ;
                a := a+(m+1)*k+l*m*(m+1)/2 ;
            end if;
        end do:
    end do:
    a ;
end proc: # R. J. Mathar, Oct 17 2012

Mathematica

A106846[n_] := Module[{a, k, l, m }, a = 0; For[k = 0, k <= n, k++, For[l = 0, l <= n, l++, If[l == 0, m = n, m = Floor[(n - k)/l]]; If[m >= 0, m = Min[m, n]; a = a + (m + 1)*k + l*m*(m + 1)/2 ]]]; a];
Table[A106846[n], {n, 0, 40}] (* Jean-François Alcover, Apr 04 2024, after R. J. Mathar *)

Prog

(Python)
from math import isqrt
def A106846(n): return (m:=n*(n+1))*((n<<1)+1-(s:=isqrt(n))**2)+s**2*(s-1)*(s+1)**2*(s+2)//9+sum(-(q:=n//k)*(k*(q+1)*(k*((q<<1)+1)-3)-6*m) for k in range(1, s+1))//3>>1 # Chai Wah Wu, Feb 02 2026

Crossrefs

Cf. A106633, A106634, A106847.

Cf. A055112, A006218, A143127, A319085, A059270, A143127.

Sequence in context: A130015 A189953 A174814 * A086863 A241689 A052149

Adjacent sequences: A106843 A106844 A106845 * A106847 A106848 A106849

Keyword

nonn

Author

Ralf Stephan, May 06 2005

Status

approved