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A300428
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).
3
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, 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, 29, 9, 19, 11, 13, 17, 25, 33, 65, 1, 33, 23, 17, 13, 11, 29, 9, 23, 7
OFFSET
1,3
COMMENTS
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.
This sequence is similar to A035335 for the base 2.
All terms are odd.
All terms appears infinitely many times (as a(n) equals at least a(2*n) or a(2*n + 1)).
See also A300475 for a similar sequence.
LINKS
FORMULA
a(2^k) = 1 for any k >= 0.
a(2^k - 1) = 2^k + 1 for any k > 1.
a(A000975(k)) = 3 for any k > 2.
a(A033138(k)) = 7 for any k > 4.
a(n) <= A300475(n) for any n > 0.
EXAMPLE
The first terms, alongside the binary representation of 1/a(n) with the earliest occurrence of the binary representation of n in parentheses, are:
n a(n) bin(1/a(n))
-- ---- -----------
1 1 (1).000...
2 1 (1.0)000...
3 5 0.00(11)001...
4 1 (1.00)000...
5 3 0.0(101)010...
6 5 0.00(110)011...
7 9 0.000(111)000...
8 1 (1.000)000...
9 5 0.001(1001)100...
10 3 0.0(1010)101...
11 11 0.000(1011)101...
12 5 0.00(1100)110...
13 11 0.000101(1101)000...
14 9 0.000(1110)001...
15 17 0.0000(1111)000...
16 1 (1.0000)000...
17 9 0.00011(10001)110...
18 7 0.00(10010)010...
19 5 0.001(10011)001...
20 11 0.0001011(10100)010...
PROG
(PARI) See Links section.
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Rémy Sigrist, Mar 05 2018
STATUS
approved