ln(m)'s close to integers

[Leroy Quet]

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,