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

[sven-h.simon]

Hello,
sorry for my late reply, but I have to relax sometimes too.

In the examples of sofpr(n) in complex plane from Mathematica, as Joseph 
Biberstine wrote, I do not understand, how sofpr(10) can be 4+5i.
10 = (-1)*(1+i)(1+i)*(1+2i)*(2+i)
When -1 is not counted as prime factor, the sum of prime factors would be 
(1+i)+(1+i)+(2+i)+(1+2i)= 5+5i. So I think Mathematica counts -1 as prime 
factor. 
What would be sofpr(100)? 100 = (1+i)**4*(1+2i)**2*(2+i)**2. The sum would be 
4*(1+i)+2*(1+2i)+2*(2+i) = 10+10i.
With the definition Mathematica uses, sofpr(10*10) <> sofpr(10)+sofpr(10).
Unfortunatly I do not have Mathematica, so I can not examine, how it calculates 
the function. 

An important property of sofpr in natural numbers is the fact that sofpr(m*n) = 
sofpr(m)+sofpr(n). This can be reached in compex plane too, when the prime 
factors are restricted to a quadrant in complex plane for example, there are 
many possible ways to do this. With such a definiton, there is only one way to 
factor a number into primes then and sofpr is well defined.

It would be possbile to use negative numbers in the factorization of natural 
numbers too, but no one does this. For example 45 = 3*3*5 = (-3)*(-3)*5 = 
(-3)*3*(-5), so we could have sofpr(45)= 11,-1 or -5. This is the same in 
complex plane, but a little more complicated. One thing that makes it more 
unpleasant is the fact, that the factors of 10 used to build sofpr are the 
factors of -10, and not exactly those of 10.

In the definition of Spira for the complex sum of divisors, published in 
American Mathematical Monthly, he uses only prime factors of the complex plane 
in the first quadrant. I used this definition too in A122435. By the way the 
definition of the sequence is not completly correct too, there are only 
composite numbers in the sequence, no primes, a prime is of course divisible by 
its sofpr anyway.

Sven