[seqfan] Re: A slightly puzzling behavior

Charles Greathouse charles.greathouse at case.edu
Sat Feb 1 04:28:19 CET 2014


For completeness, let me mention that this works with any Mersenne prime
(for sufficiently large p).

Charles Greathouse
Analyst/Programmer
Case Western Reserve University


On Fri, Jan 31, 2014 at 2:03 PM, Jack Brennen <jfb at brennen.net> 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.
>
>
>
>
> On 1/31/2014 10:39 AM, Giovanni Resta wrote:
>
>> Hi all,
>> I've computed the primitive abundant numbers (definition
>> https://oeis.org/A091191 ) up to 10^11 and I've
>> computed how many of them are divisible by the primes 2, 3, 5,... 71.
>>
>> As expected, most of them are divisible by 2 and/or 3.
>>
>> However I cannot focus on the reason why an apparently
>> anomalous fraction of them is divisible by 7 or 31
>> (both primes of the form 2^k-1). Probably it is obvious,
>> but right now I cannot see it.
>>
>> A graph is available here (the last one):
>> http://www.numbersaplenty.com/set/primitive_abundant/
>>
>> (note that the similar graph, made for abundant numbers, is
>> much more regular, showing no peak for 7 or 31).
>>
>> Any idea?
>> Giovanni
>>
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>>
>>
>>
>
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