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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.

%I A303972 #10 Sep 08 2022 08:46:21
%S A303972 0,0,27,64,125,432,343,1024,1458,2000,1331,6912,2197,5488,10125,12288,
%T A303972 4913,23328,6859,32000,27783,21296,12167,82944,31250,35152,59049,
%U A303972 87808,24389,162000,29791,131072,107811,78608,128625,326592,50653,109744,177957,384000
%N 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.
%H A303972 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A303972 a(n) = n^3 * Sum_{i=1..floor((n-1)/2)} floor((n-i)/i) - floor((n-i-1)/i).
%F A303972 a(n) = n * A303873(n).
%p A303972 A303972 := proc(n)
%p A303972     v := 0 ;
%p A303972     for p from 1 to n/2 do
%p A303972         q := n-p ;
%p A303972         if p < q and modp(q,p) = 0 then
%p A303972             v := v+n^3 ;
%p A303972         end if;
%p A303972     end do:
%p A303972     v ;
%p A303972 end proc:
%p A303972 seq(A303972(n),n=1..40) ; # _R. J. Mathar_, Jun 25 2018
%t A303972 Table[n^3*Sum[(Floor[(n - i)/i] - Floor[(n - i - 1)/i]), {i, Floor[(n - 1)/2]}], {n, 50}]
%o A303972 (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
%Y A303972 Cf. A303873, A023645 (number of contributing cubes).
%K A303972 nonn,easy
%O A303972 1,3
%A A303972 _Wesley Ivan Hurt_, May 03 2018