request for sequences: sum-of-[prime-]factors in other arithmetics
Marc LeBrun
mlb at well.com
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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