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A329735
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a(n) is the least k > 0 such that the binary representation of n appears as a substring in the binary representation of at least half of the numbers in the range 1..k.
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1
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1, 2, 14, 38, 110, 62, 1006, 2206, 5072, 21504, 7114, 3704, 13868, 4058, 4067254, 4384886, 9535340, 39157714, 20466206, 5565048, 732167206, 47755164, 24722194, 12837030, 27081364, 14017192, 231845728, 15111866, 32273342, 16292028, 17478178355102
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OFFSET
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1,2
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COMMENTS
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The sequence is well defined as for any n > 0, the proportion of numbers in the range 1..k whose binary representation contains that of n tends to 1 as k tends to infinity.
For any n > 0, the binary representation of n appears as a substring in the binary representation of a(n).
Apparently, records occur at indices n such that the representation of n in base 2^w contains only the digit 2^k for some w and k such that 0 <= k < w (see A330220).
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LINKS
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EXAMPLE
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For n = 3:
- the binary representation of 3 is "11",
- the binary representation of the first numbers, alongside the proportion p of those containing "11", is:
k bin(k) p
-- ------ ----
1 1 0
2 10 0
3 11 1/3
4 100 1/4
5 101 1/5
6 110 1/3
7 111 3/7
8 1000 3/8
9 1001 1/3
10 1010 3/10
11 1011 4/11
12 1100 5/12
13 1101 6/13
14 1110 1/2
- we first reach a proportion p >= 1/2 for k = 14,
- hence a(3) = 14.
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PROG
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(PARI) See Links section.
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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