One of the things to learn about Mathematica (Wolfram) is to search the documentation for functions that may simplify your work. Try guide/NumberTheoreticFunctions in the documentaion. There you'll find Divisors[] and DivisorSum[]. A number is perfect if Total[Divisors[n]]/2==n or DivisorSum[n, Identity]/2 == n.

It is likely that at that value the internal automatic compilation of some of the code is kicking in:

In[16]:= SystemOptions[CompileOptions]

Out[16]= {"CompileOptions" -> {"ApplyCompileLength" -> \[Infinity], 
   "ArrayCompileLength" -> 250, "AutoCompileAllowCoercion" -> False, 
   "AutoCompileProtectValues" -> False, "AutomaticCompile" -> False, 
   "BinaryTensorArithmetic" -> False, "CompileAllowCoercion" -> True, 
   "CompileConfirmInitializedVariables" -> True, 
   "CompiledFunctionArgumentCoercionTolerance" -> 2.10721, 
   "CompiledFunctionMaxFailures" -> 3, 
   "CompileDynamicScoping" -> False, 
   "CompileEvaluateConstants" -> True, 
   "CompileOptimizeRegisters" -> False, 
   "CompileParallelizationThreshold" -> 10, 
   "CompileReportCoercion" -> False, "CompileReportExternal" -> False,
    "CompileReportFailure" -> False, "CompileValuesLast" -> True, 
   "FoldCompileLength" -> 100, "InternalCompileMessages" -> False, 
   "ListableFunctionCompileLength" -> 250, "MapCompileLength" -> 100, 
   "NestCompileLength" -> 100, "NumericalAllowExternal" -> False, 
   "ProductCompileLength" -> 250, "ReuseTensorRegisters" -> True, 
   "SumCompileLength" -> 250, "SystemCompileOptimizations" -> All, 
   "TableCompileLength" -> 250}}

If you play with some of these using

SetSystemOptions[
 "CompileOptions" -> {"ApplyCompileLength" -> \[Infinity], 
   "ArrayCompileLength" -> 250, "AutoCompileAllowCoercion" -> False, 
   "AutoCompileProtectValues" -> False, "AutomaticCompile" -> False, 
   "BinaryTensorArithmetic" -> False, "CompileAllowCoercion" -> True, 
   "CompileConfirmInitializedVariables" -> True, 
   "CompiledFunctionArgumentCoercionTolerance" -> 2.1072099696478683`,
    "CompiledFunctionMaxFailures" -> 3, 
   "CompileDynamicScoping" -> False, 
   "CompileEvaluateConstants" -> True, 
   "CompileOptimizeRegisters" -> False, 
   "CompileParallelizationThreshold" -> 10, 
   "CompileReportCoercion" -> False, "CompileReportExternal" -> False,
    "CompileReportFailure" -> False, "CompileValuesLast" -> True, 
   "FoldCompileLength" -> 100, "InternalCompileMessages" -> False, 
   "ListableFunctionCompileLength" -> 250, "MapCompileLength" -> 100, 
   "NestCompileLength" -> 100, "NumericalAllowExternal" -> False, 
   "ProductCompileLength" -> 250, "ReuseTensorRegisters" -> True, 
   "SumCompileLength" -> 250, "SystemCompileOptimizations" -> All, 
   "TableCompileLength" -> 250}]

You will be able to see if the threshold where the timing makes a jump moves according to the change in the threshold for the system options setting.