cp's OEIS Frontend

A175413 Those positive integers n that when written in binary, the lengths of the runs of 1 are distinct and the lengths of the runs of 0's are distinct.

%I A175413 #26 Mar 14 2022 09:13:25
%S A175413 1,2,3,4,6,7,8,11,12,13,14,15,16,19,23,24,25,28,29,30,31,32,35,38,39,
%T A175413 44,47,48,49,50,52,55,56,57,59,60,61,62,63,64,67,70,71,78,79,88,92,95,
%U A175413 96,97,98,103,104,111,112,113,114,115,116,120,121,123,124,125
%N A175413 Those positive integers n that when written in binary, the lengths of the runs of 1 are distinct and the lengths of the runs of 0's are distinct.
%C A175413 A044813 contains those positive integers that when written in binary, have all run-lengths (of both 0's and 1's) distinct.
%C A175413 A175414 contains those positive integers in A175413 that are not in A044813. (A175414 contains those positive integers that when written in binary, at least one run of 0's is the same length as one run of 1's, even though all run of 0 are of distinct length and all runs of 1's are of distinct length.)
%C A175413 Also numbers whose binary expansion has all distinct runs (not necessarily run-lengths). - _Gus Wiseman_, Feb 21 2022
%H A175413 Alois P. Heinz, <a href="/A175413/b175413.txt">Table of n, a(n) for n = 1..20000</a>
%p A175413 q:= proc(n) uses ListTools; (l-> is(nops(l)=add(
%p A175413       nops(i), i={Split(`=`, l, 1)}) +add(
%p A175413       nops(i), i={Split(`=`, l, 0)})))(Bits[Split](n))
%p A175413     end:
%p A175413 select(q, [$1..200])[];  # _Alois P. Heinz_, Mar 14 2022
%t A175413 f[n_] := And@@Unequal@@@Transpose[Partition[Length/@Split[IntegerDigits[n, 2]], 2, 2, {1,1}, 0]]; Select[Range[125], f] (* _Ray Chandler_, Oct 21 2011 *)
%t A175413 Select[Range[0,100],UnsameQ@@Split[IntegerDigits[#,2]]&] (* _Gus Wiseman_, Feb 21 2022 *)
%o A175413 (Python)
%o A175413 from itertools import groupby, product
%o A175413 def ok(n):
%o A175413     runs = [(k, len(list(g))) for k, g in groupby(bin(n)[2:])]
%o A175413     return len(runs) == len(set(runs))
%o A175413 print([k for k in range(1, 125) if ok(k)]) # _Michael S. Branicky_, Feb 22 2022
%Y A175413 Cf. A044813, A175414.
%Y A175413 Runs in binary expansion are counted by A005811, distinct A297770.
%Y A175413 The complement is A351205.
%Y A175413 The version for standard compositions is A351290, complement A351291.
%Y A175413 A000120 counts binary weight.
%Y A175413 A242882 counts compositions with distinct multiplicities.
%Y A175413 A318928 gives runs-resistance of binary expansion.
%Y A175413 A325545 counts compositions with distinct differences.
%Y A175413 A333489 ranks anti-runs, complement A348612, counted by A003242.
%Y A175413 A334028 counts distinct parts in standard compositions.
%Y A175413 A351014 counts distinct runs in standard compositions.
%Y A175413 Counting words with all distinct runs:
%Y A175413 - A351013 = compositions, for run-lengths A329739.
%Y A175413 - A351016 = binary words, for run-lengths A351017.
%Y A175413 - A351018 = binary expansions, for run-lengths A032020.
%Y A175413 - A351200 = patterns, for run-lengths A351292.
%Y A175413 - A351202 = permutations of prime factors.
%Y A175413 Cf. A070939, A085207, A098859, A233564, A238130 or A238279, A283353, A328592, A350952, A351015, A351203.
%K A175413 nonn,base
%O A175413 1,2
%A A175413 _Leroy Quet_, May 07 2010
%E A175413 Extended by _Ray Chandler_, Oct 21 2011