sigma = DivisorSigma[0, #] &;
currentmax = -\[Infinity];
res = {};
Do[
 s = sigma[v];
 If[s > currentmax,
  AppendTo[res, v];
  currentmax = s;
  ];
 If[Length[res] >= 25, Break[]]
 ,
 {v, \[Infinity]}
 ]
res