A303873 -id:A303873 - cp's OEIS frontend

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

Wesley Ivan Hurt, May 03 2018

Index entries for sequences related to partitions

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

[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

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

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

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).

A303973 Total volume of all rectangular prisms with dimensions (p,p,q) such that n = p + q, p divides q and p < q.

Original entry on oeis.org

0, 0, 2, 3, 4, 21, 6, 31, 62, 41, 10, 260, 12, 61, 372, 263, 16, 648, 18, 722, 868, 101, 22, 2292, 524, 121, 1700, 1544, 28, 3873, 30, 2135, 2964, 161, 2156, 7703, 36, 181, 4756, 6690, 40, 9051, 42, 4844, 11088, 221, 46, 18788, 2106, 5366, 10308, 7610, 52

Offset: 1

Wesley Ivan Hurt, May 03 2018

			For n =12 the  prism (p,p,q) = (1,1,11) contributes 1*1*11=11 to the volume, (2,2,10) contributes 2*2*10= 40, (3,3,9) contributes 3*3*9= 81, (4,4,8) contributes 128. The total is a(12) = 11+40+81+128 = 260.

Index entries for sequences related to partitions

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

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

A303973 := 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+p^2*q ;
                end if;
        end do:
        v ;
end proc:
seq(A303973(n),n=1..40) ; # R. J. Mathar, Jun 25 2018

Table[Sum[i^2 (n - i) (Floor[(n - i)/i] - Floor[(n - i - 1)/i]), {i, Floor[(n - 1)/2]}], {n, 100}]

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

cp's OEIS Frontend

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.

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