ClearAll[SequenceOk]
SequenceOk[n_, k_] := Module[{m = n, q, r, valid = True},
  Do[
   {q, r} = QuotientRemainder[m, k];
   If[r != 1,
    valid = False;
    Break[];
    ];
   m -= q + 1
   ,
   {k}
   ];
  If[Mod[m, k] != 0,
   valid = False
   ];
  valid
  ]
i = 1;
While[! SequenceOk[i, 5], i++]
i

i = 1;
While[! SequenceOk[i, 6], i++]
i