A325984 Lexicographically earliest sequence of distinct nonnegative terms such that for any n >= 0, the sum of digits of n in base 2 equals the sum of digits of a(n) in base 3.

0, 1, 3, 2, 9, 4, 6, 5, 27, 10, 12, 7, 18, 11, 13, 8, 81, 28, 30, 15, 36, 19, 21, 14, 54, 29, 31, 16, 33, 20, 22, 17, 243, 82, 84, 37, 90, 39, 45, 24, 108, 55, 57, 32, 63, 34, 38, 23, 162, 83, 85, 40, 87, 42, 46, 25, 91, 48, 56, 35, 58, 41, 43, 26, 729, 244

Offset: 0

Author

Rémy Sigrist, May 30 2019

Keywords

• nonn
• base

Comments

This sequence is a permutation of the nonnegative integers, with inverse A325985.

The first known fixed points are: 0, 1, 6 and 129936.

We can generalize this sequence for any pair of bases > 1, say u and v:

- let f_{u,v} be the lexicographically earliest sequence of distinct nonnegative terms such that for any n >= 0, the sum of digits of n in base u equals the sum of digits of f_{u,v}(n) in base v,

- in particular f_{2,3} = a (this sequence) and f_{3,2} = A325985,

- f_{u,v} is a permutation of the nonnegative integers, with inverse f_{v,u},

- f_{u,v}(u^k) = v^k for any k >= 0,

- f_{u,u} is the identity function,

- f_{u,v} o f_{v,w} = f_{u,w} (where o denotes function composition).

Examples

			The first terms, alongside the binary representation of n and the ternary representation of a(n), are:
  n   a(n)  bin(n)  ter(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     1       1          1
   2     3      10         10
   3     2      11          2
   4     9     100        100
   5     4     101         11
   6     6     110         20
   7     5     111         12
   8    27    1000       1000
   9    10    1001        101
  10    12    1010        110
  11     7    1011         21
  12    18    1100        200
  13    11    1101        102
  14    13    1110        111
  15     8    1111         22
  16    81   10000      10000
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..8192
• Rémy Sigrist, Colored logarithmic scatterplot of the sequence for n = 0..2^16 (where the color is function of A000120(n)).
• Rémy Sigrist, PARI program for A325984.
• Index entries for sequences that are permutations of the natural numbers.

Crossrefs

Cf. A000120, A018900, A053735, A055235, A325985 (inverse).

Programs

Mathematica

a[0] = 0; a[n_] := a[n] = Module[{s = DigitCount[n, 2, 1], k = 1}, While[! FreeQ[Array[a, n - 1], k] || Plus @@ IntegerDigits[k, 3] != s, k++]; k]; Array[a, 66, 0] (* Amiram Eldar, Jul 25 2023 *)

PARI

See Links section.

Formula

a(2^k) = 3^k for any k >= 0.

A000120(n) = A053735(a(n)).

a(A018900(k)) = A055235(k-1) for any k > 0.