n = {m} = {1};
tmp = {2};
Do[
 m++;
 While[ContainsAny[tmp, m + n],
  m++
  ];
 tmp = Join[tmp, n + m];
 AppendTo[tmp, 2 m];
 AppendTo[n, m]
 ,
 {99}
 ]
Row[Take[n, 30], ","]
Row[Take[n, {91, 100}], ","]