[seqfan] Re: A slightly puzzling behavior

[Giovanni Resta]

On 01/31/2014 08:03 PM, Jack Brennen wrote:
> I'm guessing that most of the primitive abundant numbers divisible by 7
> are of the form:
>
>    4*7*p
>
> which is primitive abundant for all primes p >= 7.
>
> And that most of the primitive abundant numbers divisible by 31
> are of the form:
>
>    16*31*p
>
> which is primitive abundant for all primes p >= 31.


I see, thanks (also to Charles).

Thus probably the unbalance with respect to surrounding
primes depends on the fact that
16*29*p is never prim. abundant because 16*29 is abundant,
so, in a sense, it is "too much" abundant.
Conversely, 16*37*p can be primitive abundant only for primes p <= 193 
because for p > 193 it is not abundant at all.

Probably admirable numbers (A111592) have a similar graph
( http://www.numbersaplenty.com/set/admirable_number/ )
more or less for the same reason.

Giovanni