A309668 a(n) is the least positive number of the form floor(2^k/n) for some k >= 0 not yet in the sequence.

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

Offset: 1

Author

Rémy Sigrist, Aug 11 2019

Keywords

• nonn
• look
• base

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.

Examples

			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...
		

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

Crossrefs

Cf. A070939, A300475, A309734 (inverse).

Programs

PARI

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