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about
28 exceptions per million even integers. This is small enough to indicate why  
a counterexample to conjecture II has not been found, but large enough to 
arouse suspicion that a counterexample may exist.

To see how these exceptions can arise, let us look at the smallest one beyond 
the range of the Stein and Stein calculation, which is $N_2(190188) = 2069$,
$N(190190) = 2095$. The prime factors of 190188 are 2,3, and 587, giving 
$f(190188) \approx 2.0034$, whereas 190190 has prime factors 2,5,7,11,13 and
19, giving an $f$ of about 2.0535. Thus it is possible with enough small
prime factors to overcome the lack of a factor of 3 and get an $f(n)$ above 2.
This can produce quite substantial differences in the values of $N_2(n)$ for
exceptional pairs, for example $N_2(5895888) = 37220$, while $N_2(5895890) =
39199$, a difference of nearly 2000.

We have seen, however, that the equation ${N_2(n) = k}$ has many
solutions for large $k$, so it may be that this acts in favor of average,
rather than exceptional, behavior, and conjecture II will prove to be
true after all.

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{\bf Acknowledgements.}
The calculations took a total of about 600 hours using two parallel computers,
a 20 processor Sequent Symmetry and a 20 processor Encore Multimax, with some
checks done using a DECstation 3100 workstation.
I would like to thank the Caltech Concurrent Supercomputer Facility
and the Advanced Computing Research Facility at Argonne National Laboratory
for the use of these machines.
%This work was sponsored in part by DOE grants
%DE-FG03-85ER25009 and DE-AC03-81ER40050.

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\line{Caltech Concurrent Computation Program\hfill}
\line{California Institute of Technology\hfill}
\line{Pasadena, CA 91125, USA\hfill}
\line{{\it E-mail:} paulc@wega.caltech.edu\hfill}

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