A307099 Positive integers k at which k/log_2(k) is at a record closeness to an integer, without actually being an integer.

3, 10, 51, 189, 227, 356, 578, 677, 996, 3389, 38997, 69096, 149462, 2208495, 3459604, 4952236, 6710605, 48098656, 81762222, 419495413

Offset

1,1

Comments

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.

Links

Table of n, a(n) for n=1..20.

Example

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...

Mathematica

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 *)

Prog

(Python)
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

Crossrefs

Cf. A001146.

Sequence in context: A009343 A352414 A341648 * A048175 A288953 A192482

Adjacent sequences: A307096 A307097 A307098 * A307100 A307101 A307102

Keyword

nonn,more

Author

Alex Costea, Mar 24 2019

Status

approved