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

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

Author

Wesley Ivan Hurt, May 03 2018

Keywords

• nonn
• easy

Examples

			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.
		

Links

• Index entries for sequences related to partitions

Crossrefs

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

Programs

Magma

[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

Maple

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

Mathematica

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

Formula

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