A303972 Total volume of all cubes with side length n which can be split such that n = p + q, p divides q and p < q.

0, 0, 27, 64, 125, 432, 343, 1024, 1458, 2000, 1331, 6912, 2197, 5488, 10125, 12288, 4913, 23328, 6859, 32000, 27783, 21296, 12167, 82944, 31250, 35152, 59049, 87808, 24389, 162000, 29791, 131072, 107811, 78608, 128625, 326592, 50653, 109744, 177957, 384000

Offset: 1

Author

Wesley Ivan Hurt, May 03 2018

Keywords

• nonn
• easy

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A303873, A023645 (number of contributing cubes).

Programs

Magma

[0, 0] cat [&+[(((n-k) div k)-(n-k-1) div k)*n^3: k in [1..(n-1) div 2]]: n in [3..80]]; // Vincenzo Librandi, May 04 2018

Maple

A303972 := proc(n)
    v := 0 ;
    for p from 1 to n/2 do
        q := n-p ;
        if p < q and modp(q,p) = 0 then
            v := v+n^3 ;
        end if;
    end do:
    v ;
end proc:
seq(A303972(n),n=1..40) ; # R. J. Mathar, Jun 25 2018

Mathematica

Table[n^3*Sum[(Floor[(n - i)/i] - Floor[(n - i - 1)/i]), {i, Floor[(n - 1)/2]}], {n, 50}]

Formula

a(n) = n^3 * Sum_{i=1..floor((n-1)/2)} floor((n-i)/i) - floor((n-i-1)/i).

a(n) = n * A303873(n).