ClearAll[WeirdNumberQ, HasSumQ]
HasSumQ[n_Integer, xs_List] := HasSumHelperQ[n, ReverseSort[xs]]
HasSumHelperQ[n_Integer, xs_List] := Module[{h, t},
  If[Length[xs] > 0,
   h = First[xs];
   t = Drop[xs, 1];
   If[n < h,
    HasSumHelperQ[n, t]
    ,
    n == h \[Or] HasSumHelperQ[n - h, t] \[Or] HasSumHelperQ[n, t]
    ]
   ,
   False
   ]
  ]
WeirdNumberQ[n_Integer] := Module[{divs},
  divs = Most[Divisors[n]];
  If[Total[divs] > n,
   ! HasSumQ[n, divs]
   ,
   False
   ]
  ]
r = {};
n = 0;
While[
 Length[r] < 25,
 If[WeirdNumberQ[++n], AppendTo[r, n]]
 ]
Print[r]