ClearAll[PowerfulNumberQ, StrongAchillesNumberQ]
PowerfulNumberQ[n_Integer] := AllTrue[FactorInteger[n][[All, 2]], GreaterEqualThan[2]]
AchillesNumberQ[n_Integer] := Module[{divs},
  If[PowerfulNumberQ[n],
   divs = Divisors[n];
   If[Length[divs] > 2,
    divs = divs[[2 ;; -2]];
    !AnyTrue[Log[#, n] & /@ divs, IntegerQ]
    ,
    True
    ]
   ,
   False
   ]
  ]
StrongAchillesNumberQ[n_] := AchillesNumberQ[n] \[And] AchillesNumberQ[EulerPhi[n]]

n = 0;
i = 0;
Reap[While[n < 50,
   i++;
   If[AchillesNumberQ[i], n++; Sow[i]]
   ]][[2, 1]]

n = 0;
i = 0;
Reap[While[n < 20,
   i++;
   If[StrongAchillesNumberQ[i], n++; Sow[i]]
   ]][[2, 1]]

Tally[IntegerLength /@ Select[Range[9999999], AchillesNumberQ]] // Grid