# [seqfan] Re: Are sequences like A015762 finite?

Charles Greathouse charles.greathouse at case.edu
Mon Aug 21 23:47:27 CEST 2017

```With PARI/GP you can do

forfactored(n=1,10^8, if(sigma(n[2],4)%eulerphi(n[2])==0, print1(n[1]", ")))

to check up to 100 million in ~90 seconds thanks to the new forfactored
command.

If phi(n) was a random number around n, you'd expect the sequence to be
infinite. It's smaller than n (+) and like sigma_4, usually fairly smooth
(+), but sigma_4 and phi probably play poorly together at each prime (-).
If someone has a decent heuristic let me know.

Charles Greathouse
Case Western Reserve University

On Mon, Aug 21, 2017 at 4:20 PM, Harvey P. Dale <hpd at hpdale.org> wrote:

> Alonso:
>         There are no further terms up to 3 million.
>         Best,
>         Harvey
>
>
> -----Original Message-----
> From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Alonso
> Del Arte
> Sent: Monday, August 21, 2017 11:44 AM
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Subject: [seqfan] Are sequences like A015762 finite?
>
> A015762 consists of numbers n such that phi(n) divides sigma_4(n). These
> are the numbers listed:
>
> 1, 2, 3, 6, 249, 498
>
> Harvey's 2012 Mathematica program only goes up to 500, but can easily be
> changed to go much higher than that. I've taken it up to 10000 and found no
> further terms.
>
> Intuition is no substitute for proof, but no proof suggests itself to me
> at the moment. My intuition tells me that this sequence is finite.
>
> There are very similar sequences in its immediate neighborhood of
> A-numbers.
>
> Al
>
> --
> Alonso del Arte
> Author at SmashWords.com
> <https://www.smashwords.com/profile/view/AlonsoDelarte>
> Musician at ReverbNation.com <http://www.reverbnation.com/alonsodelarte>
>
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>
```