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%I #6 Jul 02 2018 16:20:12
%S 1,2,4,8,16,8,4,4,4,16,8,8,8,8,8,2,8,4,8,4,4,4,4,4,4,4,16,16,4,16,8,
%T 16,4,4,16,4,4,2,4,16,8,16,16,4,8,8,4,8,8,16,8,8,32,32,4,32,4,4,8,4,2,
%U 32,2,8,4,8,4,8,8,8,8,2,8,8,8,32,32,8,4,8,8,4,8,8,8,8,8,32,8,8,2,4,2,4,8
%N Maximal power of 2 arising when A000005 is applied repeatedly to n!.
%C Unlike the iteration of EulerPhi(A000005) or Cototient(A051953) functions, here the emerging powers of 2 are not accumulated at the terminal phase of iteration sequence. Non-2-powers can be intercalated.
%e n=53, the iterates are {53!,16174080000,840,32,6,4,3,2}, so a(53)=32, n=130, the iterates are {130!,287298761874053529600,38016,64,7,2}, so a(130)=64, n=563, the iterates are {563!,2875041108020454013464609906430286933482949481627276804096000000000, 77051520,512,10,4,3,2}, so a(563)=512.
%t Join[{1,2},Table[SelectFirst[Rest[NestWhileList[DivisorSigma[0,#]&,n!,#>2&]],IntegerQ[Log[2,#]]&],{n,3,100}]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Jul 02 2018 *)
%Y A000005, A000142, A000010, A051953, A036450-A036459.
%K nonn
%O 1,2
%A _Labos Elemer_, Feb 24 2000