[seqfan] Re: A slightly puzzling behavior

[Jack Brennen]

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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