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A307099
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Positive integers k at which k/log_2(k) is at a record closeness to an integer, without actually being an integer.
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0
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3, 10, 51, 189, 227, 356, 578, 677, 996, 3389, 38997, 69096, 149462, 2208495, 3459604, 4952236, 6710605, 48098656, 81762222, 419495413
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OFFSET
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1,1
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COMMENTS
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The closeness of a real number x to an integer is measured as abs(x-round(x)).
k/log_2(k) can also be written as log(k,2^k). Thus, this is also where 2^k is at a record closeness to a power of k (logarithmically).
k/log_2(k) is an integer iff k is in A001146, so these integers are ignored.
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LINKS
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EXAMPLE
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10/log_2(10) = 3.010... ~ 3, which is an integer. Or, 2^10 = 1024, which is close to 1000 = 10^3.
996/log_2(996) = 99.99998060...
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MATHEMATICA
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With[{s = {-1}~Join~Array[-Abs[Round[#] - #] &[#/Log2[#]] /. 0 -> -1 &, 10^5, 2]}, Rest@ Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* Michael De Vlieger, Mar 27 2019 *)
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PROG
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(Python 3)
from math import floor, ceil, log
x=2
mindif=1
while True:
logn=x/log(x, 2)
dif=min(logn-floor(logn), ceil(logn)-logn)
if dif!=0 and mindif>dif:
mindif=dif
print(x, end=", ")
x+=1
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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