A319268 Lexicographically earliest sequence of distinct positive terms such that for any n > 0, the binary representation of n^2 starts with the binary representation of a(n).

1, 2, 4, 8, 3, 9, 6, 16, 5, 12, 7, 18, 10, 24, 14, 32, 36, 20, 11, 25, 13, 15, 33, 72, 19, 21, 22, 49, 26, 28, 30, 64, 17, 144, 38, 40, 42, 45, 23, 50, 52, 27, 57, 60, 31, 66, 34, 288, 37, 39, 81, 84, 43, 91, 47, 98, 101, 105, 54, 56, 29, 120, 62, 128, 132, 68

Offset: 1

Author

Rémy Sigrist, Sep 16 2018

Keywords

• nonn
• base

Comments

This sequence is a permutation of the natural numbers with inverse A319499.

We can build a variant of this sequence for any base b > 1.

We can build a variant of this sequence for any strictly increasing sequence of nonnegative integers.

Examples

			The first terms, alongside the binary representation of n^2 with a(n) in parentheses, are:
  n   a(n)  bin(n^2)
  --  ----  --------
   1     1          (1)
   2     2        (10)0
   3     4       (100)1
   4     8      (1000)0
   5     3      (11)001
   6     9     (1001)00
   7     6     (110)001
   8    16    (10000)00
   9     5    (101)0001
  10    12    (1100)100
  11     7    (111)1001
  12    18   (10010)000
  13    10   (1010)1001
  14    24   (11000)100
  15    14   (1110)0001
  16    32  (100000)000
  17    36  (100100)001
  18    20  (10100)0100
  19    11  (1011)01001
  20    25  (11001)0000
		

Links

• Ivan Neretin, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, PARI program for A319268
• Rémy Sigrist, Colored logarithmic scatterplot of the first 10000 terms (where the color is function of A070939(n^2) - A070939(a(n)))
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A000290, A070939, A272679, A319499.

Programs

Mathematica

a = {1}; Do[r = IntegerDigits[n^2, 2]; AppendTo[a, Min@Complement[Table[FromDigits[Take[r, k], 2], {k, Length@r}],a]], {n, 2, 66}]; a (* Ivan Neretin, Oct 24 2018 *)

PARI

See Links section.