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.
Table[t^2-x^2,{t,0,10},{x,-10,10}];
Flatten[%];
Split[Sort[%2]]
{{-100,-100},{-99,-99},{-96,-96},{-91,-91},{-84,-84},{-81,-81},{-80,-80},{-77,-77},{-75,-75},{-72,-72},{-65,-65},{-64,-64,-64,-64},{-63,-63},{-60,-60},{-56,-56},{-55,-55},{-51,-51},{-49,-49},{-48,-48,-48,-48},{-45,-45,-45,-45},{-40,-40},{-39,-39},{-36,-36,-36,-36},{-35,-35},{-33,-33},{-32,-32,-32,-32},{-28,-28},{-27,-27},{-25,-25},{-24,-24,-24,-24},{-21,-21},{-20,-20},{-19,-19},{-17,-17},{-16,-16,-16,-16},{-15,-15,-15,-15},{-13,-13},{-12,-12},{-11,-11},{-9,-9,-9,-9},{-8,-8},{-7,-7},{-5,-5},{-4,-4},{-3,-3},{-1,-1},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{1},{3,3},{4},{5,5},{7,7},{8,8},{9,9,9},{11,11},{12,12},{13,13},{15,15,15,15},{16,16,16},{17,17},{19,19},{20,20},{21,21},{24,24,24,24},{25},{27,27},{28,28},{32,32,32,32},{33,33},{35,35},{36,36,36},{39,39},{40,40},{45,45,45,45},{48,48,48,48},{49},{51,51},{55,55},{56,56},{60,60},{63,63},{64,64,64},{65,65},{72,72},{75,75},{77,77},{80,80},{81},{84,84},{91,91},{96,96},{99,99},{100}}
Number of points with volume up to n (for 1D space)
num[n_]:=Sum[Floor[n/k],{k,n}]
Table[num[n],{n,10}]
{1,3,5,8,10,14,16,20,23,27}
ListPlot[Table[num[n],{n,100}],PlotJoined->True];
Differences[list_]:=Drop[list,1]-Drop[list,-1]
Differences[Table[num[n],{n,20}]]
{2,2,3,2,4,2,4,3,4,2,6,2,4,4,5,2,6,2,6}
ListPlot[Differences[Table[num[n],{n,100}]],PlotJoined->True];
For 2D space:
num2[n_]:=Sum[Floor[n/j/k],{k,n},{j,n/k}]
Table[num2[n],{n,10}]
{1,4,7,13,16,25,28,38,44,53}
ListPlot[Differences[Table[num2[n],{n,100}]],PlotJoined->True];
ListPlot[Table[num2[n],{n,100}],PlotJoined->True];
ListPlot[Table[num2[n]/n,{n,100}],PlotJoined->True];
ListPlot[Table[num[n],{n,100}],PlotJoined->True];
ListPlot[Table[num[n]/n,{n,100}],PlotJoined->True];
ListPlot[Table[num2[n]/num[n],{n,100}],PlotJoined->True];
ListPlot[Table[num2[n],{n,100}],PlotJoined->True];
For d-dimensional space:
num2[n_]:=Sum[Floor[n/j/k],{k,n},{j,n/k}]
ListPlot[Table[num[n]^(1/2),{n,100}],PlotJoined->True];