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%I #17 Aug 16 2019 15:14:41
%S 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,
%T 37,36,35,34,33,32,31,30,29,28,27,26,52,25,49,48,47,46,45,44,43,85,41,
%U 40,80,78,38,75,74,73,71,70,69,68,67,66,65,64,63,62,61
%N a(n) is the least positive number of the form floor(2^k/n) for some k >= 0 not yet in the sequence.
%C 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.
%C The sequence is a permutation of the natural numbers, with inverse A309734:
%C - for any m > 0, floor(2^k/A300475(m)) = m for some k,
%C - also, for any u > 0, floor(2^(k-u)/(A300475(m)*2^u)) = m,
%C - so the set S_m = { v such that floor(2^k/v) = m for some k >= 0 } is infinite
%C - and eventually a(n) = m for some n in S_m, QED.
%H Rémy Sigrist, <a href="/A309668/b309668.txt">Table of n, a(n) for n = 1..10000</a>
%H Rémy Sigrist, <a href="/A309668/a309668.png">Colored representation of the first 2^16 terms</a> (where the color is function of the parity of A070939(n) - A070939(a(n)))
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e The first terms, alongside the binary representations of a(n) and of 1/n (with that of a(n) in parentheses), are:
%e -- ---- --------- ---------------------
%e 1 1 1 (1).00000000000000...
%e 2 2 10 0.(10)000000000000...
%e 3 5 101 0.0(101)0101010101...
%e 4 4 100 0.0(100)0000000000...
%e 5 3 11 0.00(11)0011001100...
%e 6 10 1010 0.00(1010)10101010...
%e 7 9 1001 0.00(1001)00100100...
%e 8 8 1000 0.00(1000)00000000...
%e 9 7 111 0.000(111)00011100...
%e 10 6 110 0.000(110)01100110...
%e 11 11 1011 0.000(1011)1010001...
%e 12 21 10101 0.000(10101)010101...
%e 13 19 10011 0.000(10011)101100...
%e 14 18 10010 0.000(10010)010010...
%e 15 17 10001 0.000(10001)000100...
%o (PARI) s=1; for (n=1, 67, q=1/n; while (bittest(s, f=floor(q)), q*=2); print1 (f ", "); s+=2^f)
%Y Cf. A070939, A300475, A309734 (inverse).
%K nonn,look,base
%O 1,2
%A _Rémy Sigrist_, Aug 11 2019
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