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a(n) is the least positive k such that the binary representation of n appears as a substring in the binary representation of 1/k (ignoring the radix point and adding trailing zeros if necessary in case of a terminating expansion).
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%I #20 Mar 11 2018 10:09:06

%S 1,1,5,1,3,5,9,1,5,3,11,5,11,9,17,1,9,7,5,11,3,13,11,9,5,11,13,9,11,

%T 17,33,1,17,11,9,7,19,5,13,11,29,3,19,13,27,11,19,17,9,19,5,11,19,13,

%U 29,9,19,11,13,17,25,33,65,1,33,23,17,13,11,29,9,23,7

%N a(n) is the least positive k such that the binary representation of n appears as a substring in the binary representation of 1/k (ignoring the radix point 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) mod A062383(n) = n for some integer i >= 0.

%C This sequence is similar to A035335 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 A300475 for a similar sequence.

%H Rémy Sigrist, <a href="/A300428/b300428.txt">Table of n, a(n) for n = 1..8191</a>

%H Rémy Sigrist, <a href="/A300428/a300428.gp.txt">PARI program for A300428</a>

%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) <= A300475(n) for any n > 0.

%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 5 0.001(1001)100...

%e 10 3 0.0(1010)101...

%e 11 11 0.000(1011)101...

%e 12 5 0.00(1100)110...

%e 13 11 0.000101(1101)000...

%e 14 9 0.000(1110)001...

%e 15 17 0.0000(1111)000...

%e 16 1 (1.0000)000...

%e 17 9 0.00011(10001)110...

%e 18 7 0.00(10010)010...

%e 19 5 0.001(10011)001...

%e 20 11 0.0001011(10100)010...

%o (PARI) See Links section.

%Y Cf. A000975, A033138, A035335, A062383, A300475.

%K nonn,base

%O 1,3

%A _Rémy Sigrist_, Mar 05 2018