Wolfram Cloud Document

In general, we are interested in the set of points for which Product[a[i],{i,d}]<n, where the a[i] are sides of a cuboid.

np[n_]:=Sum[Floor[n/k],{k,n}]

np2[n_]:=Sum[Floor[n/j/k],{k,n},{j,n/k}]

These give the number of points that are equivalent under relativistic transformations to cubes with sides of lengths less than r.

Table[num[r^2],{r,10}]

{1,8,23,50,87,140,201,280,373,482}

Table[{r^2,num[r^2]},{r,10}]

{{1,1},{4,8},{9,23},{16,50},{25,87},{36,140},{49,201},{64,280},{81,373},{100,482}}

ListPlot[%,PlotJoined->True];

Table[num2[r^3],{r,10}]

{1,38,228,796,1952,4122,7501,12599,19776,29425}

ListPlot[%47/Range[Length[%47]]^2,PlotJoined->True];

ListPlot[%48/Range[Length[%48]]^3,PlotJoined->True];