[SeqFan] BinSYS Project. Was: request for sequences: sum-of-[prime-]factors in other arithmetics

[Antti Karttunen]

Marc LeBrun wrote:

> This is a request for somebody out there to compute various instances 
> of an interesting family of sequences and submit them.
>
> Usually I prefer the rule "to suggest is to volunteer", and avoid just 
> sketching proposals while leaving the work for someone else to 
> do...but if I don't in this case I'm afraid I'll forget the following 
> idea before I have enough time to work on it.  So here's the background:
>
>
> Recently there was some discussion of sequences involving 
> sum-of-[prime-]factors functions ("sofr", "sopfr"--A008472, 
> A001414--and a host of others involving these).
>
> I note that this concept has analogs in arithmetics that support 
> factorization, such as the Gaussian integers, GF(2), etc.

Please look at this:
http://szdg.lpds.sztaki.hu/szdg/desc_numsys_es.php

I bet there lurks many, many interesting families of sequences waiting 
for their submission
to OEIS! (And applications for the "rebase" -notation.)

Yours,

Antti

>
> Of course in general the factor-sums might not be integers, hence not 
> viable for the OEIS.  However with suitably artful definitions 
> integrality can sometimes be arranged.
>
> For instance, if we define the "principal" complex Gaussian factors to 
> be the conjugate pairs nearest the positive real axis, then the 
> sum-of-factors will always be a real integer because the imaginary 
> parts cancel.
>
> For example the Gaussian analogs of sofr(5)=sopfr(5) derive from 
> "principal" prime divisors 2+i, 2-i and 5, which sum to 9.
>
> In contrast, in GF(2), 5 is the square of 3 (via the usual "numbral" 
> binary encoding) so the analog of sofr(5) is 3 while the analog of 
> sopfr(5) is 0, summing with XOR as usual in GF(2).
>
> Thus many existing sequences involving sums of factors may have 
> interesting Gaussian, Eisenstein, GF(2) and other analogs.
>
>
> I think it would be neat if someone could compute and submit a bunch 
> of these analog sequences.
>
> Please let me know if you follow up on this suggestion (I *do* at 
> least have time to admire your work!)
>
>
>