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A165413 a(n) is the number of distinct lengths of runs in the binary representation of n.

%I A165413 #30 Jan 04 2021 17:03:55
%S A165413 1,1,1,2,1,2,1,2,2,1,2,1,2,2,1,2,2,2,2,2,1,2,2,2,2,2,2,2,2,2,1,2,2,2,
%T A165413 3,2,2,2,3,2,2,1,2,2,2,2,2,2,3,2,1,2,2,2,3,1,3,2,3,2,2,2,1,2,2,2,3,3,
%U A165413 2,3,2,3,2,2,2,2,2,3,3,2,2,2,2,2,1,2,2,3,2,2,2,3,2,2,2,2,3,3,2,2,2,2,2,3,2
%N A165413 a(n) is the number of distinct lengths of runs in the binary representation of n.
%C A165413 Least k whose value is n: 1, 4, 35, 536, 16775, 1060976, ..., = A165933. - _Robert G. Wilson v_, Sep 30 2009
%H A165413 Reinhard Zumkeller, <a href="/A165413/b165413.txt">Table of n, a(n) for n = 1..10000</a>
%F A165413 a(n) = 1 for n in A140690. - _Robert G. Wilson v_, Sep 30 2009
%e A165413 92 in binary is 1011100. There is a run of one 1, followed by a run of one 0, then a run of three 1's, then finally a run of two 0's. The run lengths are therefore (1,1,3,2). The distinct values of these run lengths are (1,3,2). Since there are 3 distinct values, then a(92) = 3.
%t A165413 f[n_] := Length@ Union@ Map[ Length, Split@ IntegerDigits[n, 2]]; Array[f, 105] (* _Robert G. Wilson v_, Sep 30 2009 *)
%o A165413 (Haskell)
%o A165413 import Data.List (group, nub)
%o A165413 a165413 = length . nub . map length . group . a030308_row
%o A165413 -- _Reinhard Zumkeller_, Mar 02 2013
%o A165413 (PARI)
%o A165413 binruns(n) = {
%o A165413   if (n == 0, return([1, 0]));
%o A165413   my(bag = List(), v=0);
%o A165413   while(n != 0,
%o A165413         v = valuation(n,2); listput(bag, v); n >>= v; n++;
%o A165413         v = valuation(n,2); listput(bag, v); n >>= v; n--);
%o A165413   return(Vec(bag));
%o A165413 };
%o A165413 a(n) = #Set(select(k->k, binruns(n)));
%o A165413 vector(105, i, a(i))  \\ _Gheorghe Coserea_, Sep 17 2015
%o A165413 (Python)
%o A165413 from itertools import groupby
%o A165413 def a(n): return len(set([len(list(g)) for k, g in groupby(bin(n)[2:])]))
%o A165413 print([a(n) for n in range(1, 106)]) # _Michael S. Branicky_, Jan 04 2021
%Y A165413 Cf. A140690 (locations of 1's), A165933 (locations of new highs).
%Y A165413 Cf. A005811, A165414.
%Y A165413 Cf. A030308, A044813.
%K A165413 base,nonn
%O A165413 1,4
%A A165413 _Leroy Quet_, Sep 17 2009
%E A165413 More terms from _Robert G. Wilson v_, Sep 30 2009