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A175872 Write n in binary. Consider the 0's and 1 as a list. (*) If the new list consists entirely of 1's, then a(n) = the number of 1's, and end. Otherwise, construct a new list made up of the lengths, written in order, of the runs of similarly-valued terms from the previous list. Go to *.

%I A175872 #19 Mar 11 2014 01:32:54
%S A175872 1,2,2,2,3,2,3,2,3,4,2,1,2,2,4,2,3,2,2,2,5,2,2,2,2,2,3,2,2,2,5,2,3,2,
%T A175872 3,4,2,3,3,2,2,6,2,1,3,2,2,2,3,1,1,3,2,4,3,1,3,2,3,2,2,2,6,2,3,2,3,4,
%U A175872 2,4,2,4,5,2,2,2,2,4,3,2,2,3,2,2,7,2,2,2,2,3,2,2,3,2,2,2,3,2,3,2,2,2,2,3,2,2,3,2,2
%N A175872 Write n in binary. Consider the 0's and 1 as a list. (*) If the new list consists entirely of 1's, then a(n) = the number of 1's, and end. Otherwise, construct a new list made up of the lengths, written in order, of the runs of similarly-valued terms from the previous list. Go to *.
%C A175872 n appears for the first time at A000975(n). - Sean A. Irvine and _N. J. A. Sloane_, Dec 02 2010
%C A175872 A217921(n) = number of steps needed to calculate a(n). - _Reinhard Zumkeller_, Mar 26 2013
%H A175872 Sean A. Irvine, <a href="/A175872/b175872.txt">Table of n, a(n) for n = 1..100000</a>
%e A175872 100 (decimal) in binary is 1100100. The lengths of the runs are: 2,2,1,2. The lengths of the runs in the latest list are: 2,1,1. The lengths of the runs in the latest list are: 1,2. The lengths of the runs in the latest list are: 1,1. This last list consists entirely of 1's. There are two 1's, so a(100) = 2.
%t A175872 f[n_Integer] := IntegerDigits[n, 2]; f[nn:{1..}] := nn; f[nn_List] := Length /@ Split[nn]; a[n_] := FixedPoint[f, n] // Length; Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Nov 26 2013 *)
%o A175872 (Haskell)
%o A175872 import Data.List (group, genericLength)
%o A175872 a175872 = f . a030308_row where
%o A175872    f xs | all (== 1) xs = length xs
%o A175872         | otherwise     = f $ map genericLength $ group xs
%o A175872 -- _Reinhard Zumkeller_, Mar 26 2013
%Y A175872 Cf. A101211, A000975.
%Y A175872 Cf. A030308.
%K A175872 base,nonn,nice
%O A175872 1,2
%A A175872 _Leroy Quet_, Oct 03 2010
%E A175872 a(3) corrected by _Leroy Quet_, Oct 06 2010
%E A175872 More terms from _Sean A. Irvine_, Dec 02 2010