ClearAll[a]
a[1] = 1;
a[n_?(GreaterThan[1])] := a[n] = 1/(2 Floor[a[n - 1]] + 1 - a[n - 1])
a /@ Range[20]

ClearAll[a]
a = 1;
n = 1;
Dynamic[n]
done = False;
While[! done,
 a = 1/(2 Floor[a] + 1 - a);
 n++;
 If[a == 83116/51639,
  Print[n];
  Break[];
  ]
 ]