A306231 Lexicographically earliest sequence of distinct positive terms such that for any n > 0 and any k > 0, floor((2^k) / a(n)) AND floor((2^k) / a(n+1)) = 0 (where AND denotes the bitwise AND operator).

1, 2, 3, 6, 4, 5, 20, 8, 9, 72, 16, 7, 14, 21, 78, 32, 11, 352, 64, 10, 40, 15, 24, 12, 30, 35, 390, 48, 96, 51, 102, 60, 13, 832, 117, 144, 18, 168, 42, 28, 39, 180, 56, 84, 63, 70, 780, 120, 26, 128, 19, 504, 36, 288, 126, 45, 112, 151, 896, 156, 720, 224

Offset: 1

Author

Rémy Sigrist, Jan 30 2019

Keywords

• nonn
• base

Comments

In other words, for any n > 0, the binary expansions of 1/a(n) and of 1/a(n+1) have no common one bit; in this sense, this sequence is similar to A109812.

This sequence is a permutation of the natural numbers, with inverse A306233 (we can first prove that all the powers of 2 appear in the sequence and then that every natural number appear in the sequence).

Examples

			The first terms, alongside A007733(a(n)) and the binary representation of 1/a(n) with periodic part in parentheses, are:
  n   a(n)  period  bin(1/a(n))
  --  ----  ------  -------------------
   1     1       1  1.(0)
   2     2       1  0.1(0)
   3     3       2  0.(01)
   4     6       2  0.0(01)
   5     4       1  0.01(0)
   6     5       4  0.(0011)
   7    20       4  0.00(0011)
   8     8       1  0.001(0)
   9     9       6  0.(000111)
  10    72       6  0.000(000111)
  11    16       1  0.0001(0)
  12     7       3  0.(001)
  13    14       3  0.0(001)
  14    21       6  0.(000011)
  15    78      12  0.0(000001101001)
  16    32       1  0.00001(0)
  17    11      10  0.(0001011101)
  18   352      10  0.00000(0001011101)
  19    64       1  0.000001(0)
  20    10       4  0.0(0011)
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..2000
• Rémy Sigrist, PARI program for A306231
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A000120, A007733, A109812, A306233 (inverse).

Programs

PARI

See Links section.

Formula

For any n > 0, if A000120(a(n)) <> 1 and A000120(a(n+1)) <> 1, then gcd(A007733(a(n)), A007733(a(n+1))) > 1.