palindromeQ[n_Integer] :=
 Block[{digits = IntegerDigits[n]}, digits == Reverse@digits]

nextNumber[n_Integer] := n + FromDigits[Reverse@IntegerDigits@n]

lychrelQ[n_Integer] := !
  palindromeQ@
   Catch[Nest[If[palindromeQ[#], Throw[#], nextNumber[#]] &,
     nextNumber[n], 500]]