%I #16 Mar 11 2018 17:52:08
%S 1,1,5,1,3,5,9,1,7,3,11,5,19,9,17,1,15,7,13,25,3,23,11,21,5,19,37,9,
%T 35,17,33,1,31,15,29,7,27,53,13,25,49,3,47,23,45,11,43,21,41,81,5,39,
%U 19,75,37,9,71,35,69,17,67,33,65,1,63,31,61,15,59,29,57
%N a(n) is the least positive k such that the binary representation n appears in front of the binary representation of 1/k (ignoring the radix point and the leading zeros and adding trailing zeros if necessary in case of a terminating expansion).
%C In other words, a(n) is the least k > 0 such that floor((2^i) / k) = n for some integer i >= 0.
%C This sequence is similar to A095156 for the base 2.
%C All terms are odd.
%C All terms appears infinitely many times (as a(n) equals at least a(2*n) or a(2*n + 1)).
%C See also A300428 for a similar sequence.
%H Rémy Sigrist, <a href="/A300475/b300475.txt">Table of n, a(n) for n = 1..10000</a>
%H Rémy Sigrist, <a href="/A300475/a300475.gp.txt">PARI program for A300475</a>
%H Rémy Sigrist, <a href="/A300475/a300475.png">Colored logarithmic scatterplot of the first 1000000 terms</a> (where the color is function of A070939(n * a(n)))
%F a(2^k) = 1 for any k >= 0.
%F a(2^k - 1) = 2^k + 1 for any k > 1.
%F a(A000975(k)) = 3 for any k > 2.
%F a(A033138(k)) = 7 for any k > 4.
%F a(n) >= A300428(n).
%e The first terms, alongside the binary representation of 1/a(n) with the earliest occurrence of the binary representation of n in parentheses, are:
%e n a(n) bin(1/a(n))
%e -- ---- -----------
%e 1 1 (1).000...
%e 2 1 (1.0)000...
%e 3 5 0.00(11)001...
%e 4 1 (1.00)000...
%e 5 3 0.0(101)010...
%e 6 5 0.00(110)011...
%e 7 9 0.000(111)000...
%e 8 1 (1.000)000...
%e 9 7 0.00(1001)001...
%e 10 3 0.0(1010)101...
%e 11 11 0.000(1011)101...
%e 12 5 0.00(1100)110...
%e 13 19 0.0000(1101)011...
%e 14 9 0.000(1110)001...
%e 15 17 0.0000(1111)000...
%e 16 1 (1.0000)000...
%e 17 15 0.000(10001)000...
%e 18 7 0.00(10010)010...
%e 19 13 0.000(10011)101...
%e 20 25 0.0000(10100)011...
%o (PARI) See Links section.
%Y Cf. A000975, A033138, A070939, A095156, A300428.
%K nonn,look,base
%O 1,3
%A _Rémy Sigrist_, Mar 06 2018