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)))
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
KEYWORD
AUTHOR
Rémy Sigrist, Aug 11 2019
STATUS
approved