A303973 - OEIS

%I #10 Sep 08 2022 08:46:21

%S 0,0,2,3,4,21,6,31,62,41,10,260,12,61,372,263,16,648,18,722,868,101,

%T 22,2292,524,121,1700,1544,28,3873,30,2135,2964,161,2156,7703,36,181,

%U 4756,6690,40,9051,42,4844,11088,221,46,18788,2106,5366,10308,7610,52

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

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

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

%e 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 := proc(n)

%p         v := 0 ;

%p         for p from 1 to n/2 do

%p                 q := n-p ;

%p                 if p < q and modp(q,p) = 0 then

%p                         v := v+p^2*q ;

%p                 end if;

%p         end do:

%p         v ;

%p end proc:

%p seq(A303973(n),n=1..40) ; # _R. J. Mathar_, Jun 25 2018

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

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

%K nonn,easy

%O 1,3

%A _Wesley Ivan Hurt_, May 03 2018