request for sequences: sum-of-[prime-]factors in other arithmetics

Marc LeBrun mlb at
Sat Jul 29 19:26:56 CEST 2006

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.

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!)

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