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

%I A303973 #10 Sep 08 2022 08:46:21
%S A303973 0,0,2,3,4,21,6,31,62,41,10,260,12,61,372,263,16,648,18,722,868,101,
%T A303973 22,2292,524,121,1700,1544,28,3873,30,2135,2964,161,2156,7703,36,181,
%U A303973 4756,6690,40,9051,42,4844,11088,221,46,18788,2106,5366,10308,7610,52
%N A303973 Total volume of all rectangular prisms with dimensions (p,p,q) such that n = p + q, p divides q and p < q.
%H A303973 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A303973 a(n) = Sum_{i=1..floor((n-1)/2)} i^2 * (n-i) * (floor((n-i)/i) - floor((n-i-1)/i)).
%e A303973 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.
%p A303973 A303973 := proc(n)
%p A303973         v := 0 ;
%p A303973         for p from 1 to n/2 do
%p A303973                 q := n-p ;
%p A303973                 if p < q and modp(q,p) = 0 then
%p A303973                         v := v+p^2*q ;
%p A303973                 end if;
%p A303973         end do:
%p A303973         v ;
%p A303973 end proc:
%p A303973 seq(A303973(n),n=1..40) ; # _R. J. Mathar_, Jun 25 2018
%t A303973 Table[Sum[i^2 (n - i) (Floor[(n - i)/i] - Floor[(n - i - 1)/i]), {i, Floor[(n - 1)/2]}], {n, 100}]
%o A303973 (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
%Y A303973 Cf. A303873, A023645 (number of contributing prisms).
%K A303973 nonn,easy
%O A303973 1,3
%A A303973 _Wesley Ivan Hurt_, May 03 2018