A070962

[Dean Hickerson]

franktaw at netscape.net wrote:

> I  have just sent in a correction to this sequence; the old name
> claimed it was the number of k<=n with omega(k)=omega(n); actually it
> is the number that are not equal.
>
> The entry contains the claim, if I'm reading it correctly, that lim
> inf_{n->inf} a(n)/n = 0.59....  This seems extraordinarily unlikely to
> me, since omega(n) has average order log(log(n)) as n->infinitiy.  Can
> anyone determine the source for this claim?  Verify or disprove it?

It's false; in fact lim a(n)/n = 1.  This follows from the Erdos-Kac
theorem on the distribution of values of omega(n); see

    http://mathworld.wolfram.com/Erdos-KacTheorem.html

As for the source of the claim, it's probably just based on numerical data
for n about 10000.  The sequence's author has often stated conjectures about
sequences (without even saying that they're conjectures) based only on such
data.  For sequences related to primes, such claims are frequently false.

Dean Hickerson
dean at math.ucdavis.edu