First Occurrence Of Floor(m/d(m))

[Max A.]

On 12/30/06, Dean Hickerson <dean at math.ucdavis.edu> wrote:

> > Note that d(m) <= log2(m), implying that m/d(m) >= m/log2(m) tends to
> > infinity as m grows.
>
> This is false for m=1, 2, 3, 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, ...

Yes, my mistake. Somehow I was thinking about prime divisors instead
of all divisors here.
My arguments hold for the number of (all) divisors as well but the
upper bound to d(n) must be corrected.

I've found the following quote:

: Theorem 317 of Hardy & Wright, The Theory of Numbers, says that if e > 0
: then d(n) < 2^{(1 + e) log n / log log n} for all n > n_0 (e),
: and d(n) > 2^{(1 - e) log n / log log n} for infinitely many n.
: Here, d(n) is the number of divisors of n.

Fix some e>0. Then for m>n_0(e), d(m) < 2^{(1 + e) log m / log log m},
implying that
m/d(m) > m/2^{(1 + e) log m / log log m} that tends to infinity as m grows.

Therefore, the answer to the second question is still YES, i.e., for
every fixed n, there is only a finite number of solutions to
floor(m/d(m))=n, and hence there exists a largest solution.

The first question is harder to answer. I still believe that there
exists n for which floor(m/d(m))=n has no solutions but a smallest
such n may be quite large.

Max