[seqfan] Re: A slightly puzzling behavior

Charles Greathouse charles.greathouse at case.edu
Sat Feb 1 18:22:24 CET 2014


Right, for the trick to work you need a perfect number else you get only
finitely many primitive abundants.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University


On Sat, Feb 1, 2014 at 6:15 AM, Giovanni Resta <g.resta at iit.cnr.it> wrote:

> 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
>
>
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