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

Author

Alex Costea, Mar 24 2019

Keywords

• nonn
• more

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.

Examples

			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...
		

Crossrefs

Cf. A001146.

Programs

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 *)

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