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A341910 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, the number of runs in the binary expansion of n equals the number of ones in the binary expansion of a(n).

%I A341910 #22 Feb 24 2021 16:17:57
%S A341910 0,1,3,2,5,7,6,4,9,11,15,13,10,14,12,8,17,19,23,21,27,31,29,22,18,25,
%T A341910 30,26,20,28,24,16,33,35,39,37,43,47,45,38,46,55,63,59,51,61,53,41,34,
%U A341910 42,54,44,57,62,58,49,36,50,60,52,40,56,48,32,65,67,71,69
%N A341910 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, the number of runs in the binary expansion of n equals the number of ones in the binary expansion of a(n).
%C A341910 This sequence is a permutation of the nonnegative integers with inverse A341911.
%C A341910 Apparently, A037481 corresponds to the fixed points of this sequence.
%H A341910 Rémy Sigrist, <a href="/A341910/b341910.txt">Table of n, a(n) for n = 0..8191</a>
%H A341910 Rémy Sigrist, <a href="/A341910/a341910.png">Colored scatterplot of the first 2^16 terms</a> (where the color is function of A005811(n))
%H A341910 Rémy Sigrist, <a href="/A341910/a341910.gp.txt">PARI program for A341910</a>
%H A341910 Rémy Sigrist, <a href="/A341910/a341910_1.gp.txt">PARI program for A341910 (faster)</a>
%H A341910 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H A341910 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A341910 A005811(n) = A000120(a(n)).
%F A341910 a(n) < 2^k for any n < 2^k.
%e A341910 The first terms, in decimal and in binary, are:
%e A341910   n   a(n)  bin(n)   bin(a(n))
%e A341910   --  ----  -------  ---------
%e A341910    0     0        0          0
%e A341910    1     1        1          1
%e A341910    2     3       10         11
%e A341910    3     2       11         10
%e A341910    4     5      100        101
%e A341910    5     7      101        111
%e A341910    6     6      110        110
%e A341910    7     4      111        100
%e A341910    8     9     1000       1001
%e A341910    9    11     1001       1011
%e A341910   10    15     1010       1111
%t A341910 Block[{a = {0}, k}, Do[k = 1; While[Nand[FreeQ[a, k], DigitCount[k, 2, 1] == #], k++] &@ Length[Split@ IntegerDigits[i, 2]]; AppendTo[a, k], {i, 67}]; a] (* _Michael De Vlieger_, Feb 24 2021 *)
%o A341910 (PARI) See Links section.
%Y A341910 Cf. A000120, A005811, A037481, A298847, A341911 (inverse).
%K A341910 nonn,look,base
%O A341910 0,3
%A A341910 _Rémy Sigrist_, Feb 23 2021