When (d(n+1)-d(n))*(-1)^n is positive (d(n) = A000005(n))

[Joshua Zucker]

On Sun, Mar 2, 2008 at 1:55 PM, Maximilian Hasler
<maximilian.hasler at gmail.com> wrote:
>  >  So I wonder, can we determine the limit, as n ->
>  >  infinity,  of a(n)/n?

>  yes : it decreases steadily.

We're asking for places where an even number has fewer factors than a
neighboring odd number.  Intiutively (that is, nonrigorously)
speaking, as numbers have more and more factors, things tend to
"average out" -- that is, we know the density of primes drops, but so
does the density of numbers that are products of 2 primes, or of the
form p^2 -- in general, there are fewer small values of d(n).  So,
sticking a 2 in -- which, all else being equal, doubles d(n), in the
sense that for odd k, d(2k) = 2d(k) -- is going to have an increase
which is proportionally more and more important as the numbers get
larger.

So it makes sense to me that this would be asymptotic to 0.  Though of
course I suspect that a(n) keeps increasing, just more and more
slowly.

I'll leave it to someone who knows what they're talking about to give
a more precise idea about the asymptotics of this thing.

--Joshua Zucker