A341911 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, the number of ones in the binary expansion of n equals the number of runs in the binary expansion of a(n).

0, 1, 3, 2, 7, 4, 6, 5, 15, 8, 12, 9, 14, 11, 13, 10, 31, 16, 24, 17, 28, 19, 23, 18, 30, 25, 27, 20, 29, 22, 26, 21, 63, 32, 48, 33, 56, 35, 39, 34, 60, 47, 49, 36, 51, 38, 40, 37, 62, 55, 57, 44, 59, 46, 50, 41, 61, 52, 54, 43, 58, 45, 53, 42, 127, 64, 96

Offset: 0

Author

Rémy Sigrist, Feb 23 2021

Keywords

• nonn
• look
• base

Comments

This sequence is a permutation of the nonnegative integers with inverse A341910.

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     7      100        111
   5     4      101        100
   6     6      110        110
   7     5      111        101
   8    15     1000       1111
   9     8     1001       1000
  10    12     1010       1100
		

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 A000120(n))
• Rémy Sigrist, PARI program for A341911
• 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, A298847, A341910 (inverse).

Programs

Mathematica

Block[{a = {0}, k}, Do[k = 1; While[Nand[FreeQ[a, k], Length[Split@ IntegerDigits[k, 2]] == #], k++] &@ DigitCount[i, 2, 1]; AppendTo[a, k], {i, 66}]; a] (* Michael De Vlieger, Feb 24 2021 *)

PARI

See Links section.

Formula

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

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