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).

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, 30, 26, 20, 28, 24, 16, 33, 35, 39, 37, 43, 47, 45, 38, 46, 55, 63, 59, 51, 61, 53, 41, 34, 42, 54, 44, 57, 62, 58, 49, 36, 50, 60, 52, 40, 56, 48, 32, 65, 67, 71, 69

Offset: 0

Author

Rémy Sigrist, Feb 23 2021

Keywords

• nonn
• look
• base

Comments

This sequence is a permutation of the nonnegative integers with inverse A341911.

Apparently, A037481 corresponds to the fixed points of this sequence.

Examples

			The first terms, in decimal and in binary, are:
  n   a(n)  bin(n)   bin(a(n))
  --  ----  -------  ---------
   0     0        0          0
   1     1        1          1
   2     3       10         11
   3     2       11         10
   4     5      100        101
   5     7      101        111
   6     6      110        110
   7     4      111        100
   8     9     1000       1001
   9    11     1001       1011
  10    15     1010       1111
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..8191
• Rémy Sigrist, Colored scatterplot of the first 2^16 terms (where the color is function of A005811(n))
• Rémy Sigrist, PARI program for A341910
• Rémy Sigrist, PARI program for A341910 (faster)
• Index entries for sequences related to binary expansion of n
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A000120, A005811, A037481, A298847, A341911 (inverse).

Programs

Mathematica

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 *)

PARI

See Links section.

Formula

A005811(n) = A000120(a(n)).

a(n) < 2^k for any n < 2^k.