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A140690 A positive integer n is included if n written in binary can be subdivided into a number of runs all of equal-length, the first run from the left consisting of all 1's, the next run consisting of all 0's, the next run consisting of all 1's, the next run consisting of all 0's, etc.

%I A140690 #26 Apr 07 2020 21:12:05
%S A140690 1,2,3,5,7,10,12,15,21,31,42,51,56,63,85,127,170,204,240,255,341,455,
%T A140690 511,682,819,992,1023,1365,2047,2730,3276,3640,3855,4032,4095,5461,
%U A140690 8191,10922,13107,16256,16383,21845,29127,31775,32767,43690,52428,61680
%N A140690 A positive integer n is included if n written in binary can be subdivided into a number of runs all of equal-length, the first run from the left consisting of all 1's, the next run consisting of all 0's, the next run consisting of all 1's, the next run consisting of all 0's, etc.
%C A140690 Also: numbers of the form (2^s-1)*[4^{s*(k+1)}-1]/(4^s-1) or 2^s(2^s-1)*[4^{s*(k+1)}-1]/(4^s-1), s>=1, k>=0. Subsequences are, with the possible exception of terms at n=0, A002450(n), A043291(n), A015565(2n), A093134(2n+1), A000225(n), A020522(n). [_R. J. Mathar_, Aug 04 2008]
%C A140690 From _Emeric Deutsch_, Jan 25 2018: (Start)
%C A140690 Also the indices of the compositions having equal parts.
%C A140690 We define the index of a composition to be the positive integer whose binary form has run-lengths (i.e. runs of 1's, runs of 0's, etc., from left to right) equal to the parts of the composition. Example: the composition [1,1,3,1] has index 46 since the binary form of 46 is 101110. The integer 992 is in the sequence since its binary form is 1111100000 and the composition [5,5] has equal parts. The integer 100 is not in the sequence since its binary form is 1100100 and the composition [2,2,1,2] does not have equal parts.
%C A140690 The command c(n) from the Maple program yields the composition having index n. (End)
%H A140690 Reinhard Zumkeller, <a href="/A140690/b140690.txt">Table of n, a(n) for n = 1..10000</a>
%H A140690 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%e A140690 819 in binary is 1100110011. The runs of 0's and 1's are (11)(00)(11)(00)(11). Each run (alternating 1's and 0's) is the same length. So 819 is in the sequence.
%p A140690 Runs := proc (L) local j, r, i, k: j := 1: r[j] := L[1]: for i from 2 to nops(L) do if L[i] = L[i-1] then r[j] := r[j], L[i] else j := j+1: r[j] := L[i] end if end do: [seq([r[k]], k = 1 .. j)] end proc: RunLengths := proc (L) map(nops, Runs(L)) end proc: c := proc (n) ListTools:-Reverse(convert(n, base, 2)): RunLengths(%) end proc: A := {}: for n to 62000 do if nops(convert(c(n), set)) = 1 then A := `union`(A, {n}) else  end if end do: A; # most of the Maple program is due to W. Edwin Clark. - _Emeric Deutsch_, Jan 25 2018
%t A140690 Select[Range[62000],Length[Union[Length/@Split[IntegerDigits[#,2]]]]==1&] (* _Harvey P. Dale_, Mar 22 2012 *)
%o A140690 (Haskell)
%o A140690 import Data.Set (singleton, deleteFindMin, insert)
%o A140690 a140690 n = a140690_list !! (n-1)
%o A140690 a140690_list = f $ singleton (1, 1, 2) where
%o A140690    f s | k == 1 = m : f (insert (2*b-1, 1, 2*b) $ insert (b*m, k+1, b) s')
%o A140690        | even k    = m : f (insert (b*m+b-1, k+1, b) s')
%o A140690        | otherwise = m : f (insert (b*m, k+1, b) s')
%o A140690        where ((m, k, b), s') = deleteFindMin s
%o A140690 -- _Reinhard Zumkeller_, Feb 21 2014
%Y A140690 Cf. A101211, A298644.
%K A140690 base,nonn
%O A140690 1,2
%A A140690 _Leroy Quet_, Jul 11 2008
%E A140690 Terms beyond 42 from _R. J. Mathar_, Aug 04 2008