# first Maple program to calculate the sequence.

A375786:=proc(n)
   local a,V,x,y,z;
   a:=n^3;
   for x to n do
      for y from x to floor(sqrt(x^2+3*n^2)-x) do
         z:=(3*n^2-y*x)/(x+y);
         if type(z,integer) then
            V:=y*x*z;
            if V<a then
               a:=V
            fi
         fi
      od
   od;
   return a
end proc;
seq(A375786(n),n=1..49);

# second Maple program to calculate the cuboid [x,y,z] with smallest volume for a given cube edge length n.

A375786xyz:=proc(n)
   local a,L,V,x,y,z;
   a:=n^3;
   L:=[n,n,n];
   for x to n do
      for y from x to floor(sqrt(x^2+3*n^2)-x) do
         z:=(3*n^2-y*x)/(x+y);
         if type(z,integer) then
            V:=y*x*z;
            if V<a then
               a:=V;
               L:=[x,y,z]
            fi
         fi
      od
   od;
   return L
end proc:
A375786xyz(6);