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

[Joseph Biberstine]

A late reply will be better than none.  Does this definition suit you?
It works as before, but discards any factor -1, I, or -I.

sogpfr[n_] := Total[Times @@@ Select[FactorInteger[n, GaussianIntegers
-> True], !MemberQ[{-1, I, -I}, #[[1]]] &]];

For n=2..20 it gives the following values:

{2 + 2*I, 3, 4 + 4*I, 3 + 3*I, 5 + 2*I, 7, 6 + 6*I, 6, 5 + 5*I, 11, 7 +
4*I, 5 + 5*I, 9 + 2*I, 6 + 3*I, 8 + 8*I, 5 + 5*I, 8 + 2*I, 19, 7 + 7*I}

Additionally, sogpfr[a*b]-sogpfr[a]-sogpfr[b] = 0 for all 1 <= a,b <= 200.

Does everyone agree with this amendment?  I do.

Here amended are the sequences given by my initial message.

re(sogpfr(n)):
{2, 3, 4, 3, 5, 7, 6, 6, 5, 11, 7, 5, 9, 6, 8, 5, 8, 19}
This is A78908.

im(sogpfr(n)):
{2, 0, 4, 3, 2, 0, 6, 0, 5, 0, 4, 5, 2, 3, 8, 5, 2, 0}
This is A78909.

im(sogpfr(n))>0:
{2, 4, 3, 2, 6, 5, 4, 5, 2, 3, 8, 5, 2, 7}
This is not in OEIS.

For all n there exist positive integers k,d with d squarefree such that
abs(sogpfr(n)) = k*sqrt(d).  Call k the k-part and d the squarefree part.

k-part of sogpfr(n):
{1, 1, 1, 2, 29, 1, 1, 1, 2, 1, 65, 2, 85, 5, 1}
This is not in OEIS.

k-parts exceeding 1:
{2, 29, 2, 65, 2, 85, 5, 2, 17, 2, 173, 13}
This is not in OEIS.

squarefree part of sogpfr(n):
{2, 3, 2, 3, 1, 7, 6, 6, 5, 11, 1, 5, 1, 3, 2, 5}
This is not in OEIS.

floor(abs(sogpfr(n))):
{2, 3, 5, 4, 5, 7, 8, 6, 7, 11, 8, 7, 9, 6, 11, 7, 8}
This is not in OEIS.

-JRB

sven-h.simon wrote:
> 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 
>