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A309668 a(n) is the least positive number of the form floor(2^k/n) for some k >= 0 not yet in the sequence. 2
1, 2, 5, 4, 3, 10, 9, 8, 7, 6, 11, 21, 19, 18, 17, 16, 15, 14, 13, 12, 24, 23, 22, 42, 20, 39, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 52, 25, 49, 48, 47, 46, 45, 44, 43, 85, 41, 40, 80, 78, 38, 75, 74, 73, 71, 70, 69, 68, 67, 66, 65, 64, 63, 62, 61 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The sequence is well defined as for any n > 0, there are infinitely many positive numbers of the form floor(2^k/n) with k >= 0.

The sequence is a permutation of the natural numbers, with inverse A309734:

- for any m > 0, floor(2^k/A300475(m)) = m for some k,

- also, for any u > 0, floor(2^(k-u)/(A300475(m)*2^u)) = m,

- so the set S_m = { v such that floor(2^k/v) = m for some k >= 0 } is infinite

- and eventually a(n) = m for some n in S_m, QED.

LINKS

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Rémy Sigrist, Colored representation of the first 2^16 terms (where the color is function of the parity of A070939(n) - A070939(a(n)))

Index entries for sequences that are permutations of the natural numbers

EXAMPLE

The first terms, alongside the binary representations of a(n) and of 1/n (with that of a(n) in parentheses), are:

  --  ----  ---------  ---------------------

   1     1          1  (1).00000000000000...

   2     2         10  0.(10)000000000000...

   3     5        101  0.0(101)0101010101...

   4     4        100  0.0(100)0000000000...

   5     3         11  0.00(11)0011001100...

   6    10       1010  0.00(1010)10101010...

   7     9       1001  0.00(1001)00100100...

   8     8       1000  0.00(1000)00000000...

   9     7        111  0.000(111)00011100...

  10     6        110  0.000(110)01100110...

  11    11       1011  0.000(1011)1010001...

  12    21      10101  0.000(10101)010101...

  13    19      10011  0.000(10011)101100...

  14    18      10010  0.000(10010)010010...

  15    17      10001  0.000(10001)000100...

PROG

(PARI) s=1; for (n=1, 67, q=1/n; while (bittest(s, f=floor(q)), q*=2); print1 (f ", "); s+=2^f)

CROSSREFS

Cf. A070939, A300475, A309734 (inverse).

Sequence in context: A132664 A072029 A309734 * A238758 A065652 A235200

Adjacent sequences:  A309665 A309666 A309667 * A309669 A309670 A309671

KEYWORD

nonn,look,base

AUTHOR

Rémy Sigrist, Aug 11 2019

STATUS

approved

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Last modified August 9 12:53 EDT 2020. Contains 336323 sequences. (Running on oeis4.)