countSolutions[n_]:=Length[Solve[1/x+1/y==1/n&&x>0&&y>0&&x>=y,{x,y},Integers]]

DeleteCases[Table[{n,countSolutions[n]},{n,1,100}], {_,cnt_}/;cnt<=15]

Do[cnt=countSolutions[n];If[cnt>1000,Print[{n,cnt}];Abort[]],{n,1,10^10}]