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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).
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%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