greedy computation of pseudoperfect numbers?

[N. J. A. Sloane]

While some of you have this in mind, can
you make any comment on Erd"os's problem:

Erd\H{o}s has shown that their density [i.e.,
that of the pseudoperfect numbers] exists and
says that presumably there are integers $n$
which are not pseudoperfect, but for which
$n=ab$ with $a$ abundant and $b$ having many
prime factors: can $b$ in fact have many
factors $<a$?

Thanks in anticipation of your comments (or
examples).    R.

On Tue, 8 Apr 2008, Joerg Arndt wrote:

> Thanks for the good pointer!
>
> I was just too sure no odd n would turn up.
>
> allowing divisor one, greedy gives these odd n<10000:
> 945, 1575, 2205, 2835, 3465, 4725, 5355, 5985, 6435, 6615, 6825, 7245,
> 7425, 7875, 8085, 8415, 8505, 8925, 9135, 9765,
>
> forbidding divisor one, greedy gives:
> 1575, 2835, 3465, 6435, 6615, 7245, 7425, 7875, 8085, 8415, 8505,
> 8925, 9135, 9765,
>
> From the webpage:
> "the only odd obundant numbers for which the greedy
> algorithm doesn't succeed are" (n<40000)
> 4095 5775 9555 12915 21945 22275 24255 27405 29925 35805 38745
>
> (note all divisible by 5.)
> It is not said whether these are pseudoperfect.
>
> All in all an interesting but apparently hard problem.
>
>
> * Joshua Zucker <joshua.zucker at gmail.com> [Apr 08. 2008 14:36]:
>> On Mon, Apr 7, 2008 at 9:14 PM, Joerg Arndt <arndt at jjj.de> wrote:
>>> For both sorts of pseudoperfect numbers, allowing divisor one A005835,
>>>  and forbidding divisor one A136446, a greedy method _seems_ to always
>>>  work for deciding whther a given n in in the sequence:
>>
>> I think if you scroll down at
>>   http://www.ics.uci.edu/~eppstein/numth/egypt/odd-one.html
>>
>> you'll find that it only USUALLY works, not always.
>>
>>>  Moreover, (disallowing the divisors one,) can any odd number n be in
>>>  sequence A136446?  This is of interest because all gcd computations in
>>>  a Rabin-like irreducibility test for polynomials over finite fields
>>>  (have to write it up) can be avoided for all numbers _not_ in A136446.
>>
>> In the same source it looks like there's a conjecture that there is an
>> odd number in the sequence but that nobody had crunched enough numbers
>> to find it.
>>
>> --Joshua Zucker
>
>
>