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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). 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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).

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

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.

EXAMPLE

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)

PROG

(PARI) See Links section.

CROSSREFS

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

Sequence in context: A072637 A138153 A209775 * A125703 A156688 A019567

Adjacent sequences:  A306228 A306229 A306230 * A306232 A306233 A306234

KEYWORD

nonn,base

AUTHOR

Rémy Sigrist, Jan 30 2019

STATUS

approved

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Last modified September 27 08:22 EDT 2021. Contains 347689 sequences. (Running on oeis4.)