[seqfan] Re: A conjeture about 34155

[Jens Voß]

Am 15.04.2012 00:38, schrieb Psychedelic Geometry:
> [...]
>
> The abundances of the elements of A<10^6 are {12, 56, 810, 992} , (the
> value 810 only happens for 34155)
>

Hello,

one obvious construction of elements of A is the following:

If n is perfect and q a prime > n, then nq is in A (and its abundance is 
2n). This accounts for the abundances 12, 56 and 992.

A similar construction takes into consideration that all known perfect 
numbers have the form n = 2^(p-1) * (2^p - 1):

If 2^(p-1) * (2^p - 1) is perfect, then 2^(p-1) * (2^p - 1)^3 lies in A. 
This accounts for the numbers 54 and 1372 in A.

I have not had the time to check whether there are any exceptions 
(besides 34155) to these patterns - if not, this would really make 34155 
very special!

Regards,
Jens