ln(m)'s close to integers

[Robert G. Wilson v]

Et al,

	The sequence begins:
2, 3, 7, 20, 148, 403, 1096, 1097, 2980, 2981, 8103, 59874, 162755, 442413, 
1202604, 8886110, 8886111,
I used the Mathematica coding of:
a = 1; Do[d = Abs[ N[ Round[ Log[n]] - Log[n], 24]]; If[d < a, Print[n]; a = d], 
{n, 2, 10^6}]. I am sure that there must be a better way.
16-Log(8886111)=0.0000000539...

Thanks, Bob.

Leroy Quet wrote:
> Consider the integers m, m >= 2, such that:
> ln(m) is closer to an integer than any previous term of the sequence.
> 
> I THINK (calculated much too carelessly with a hand-calculator), that
> the sequence begins:
> 2, 3, 7, 20, ....
> 
> (This matches a few sequences in the EIS, but this fact does not mean
> much, given that I only "know" 4 terms.)
> 
> I would guess that the sequence would contain consecutive terms,
> sometimes, which are close to the SAME integer.
> (If floor(ln(m)+1/2) = ceiling(ln(m)) = n, then perhaps (n-ln(m)) >
> ln(m+1) - ln(m).)
> 
> But almost anything about this sequence (such as its actual terms) is
> unknown to me at this time.
> 
> Anything to add?
> 
> Thanks,
>