[seqfan] Re: Pairs Occurring Only Once Among # Of Divisors

[Maximilian Hasler]

On Thu, Jun 25, 2009 at 6:44 PM, T. D. Noe<noe at sspectra.com> wrote:
>>> On behalf of the Quetau pairs A161640 invented in
>>> http://list.seqfan.eu/pipermail/seqfan/2009-June/001652.html
> It is also easy to prove 1023:(...)
> I'm guessing that a similar proof can be worked out for n = 2^(p-1)-1 where
> p is prime.  This proof was for p=11.

Indeed, it is also easy to prove 4095:
d(4095)=24
d(4096)=13

If n+1 has 13 divisors, then p^12 is the only possible form, with p prime.
However,
p^12 - 1 = (p-1)(p+1)(p^2-p+1)(p^2+1)(p^2+p+1)(p^4-p^2+1)
which is, for p>2, of the form
2a * 2b * c * 2d * e * f
with a,b,c,d,e,f odd numbers.
A number of this form cannot have less than 28 divisors.

So p=2 is the only possibility.

Maximilian