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

[Tautócrona]

----- Original Message ----- 
From: "Joseph Biberstine" <jrbibers at indiana.edu>


>Before I submit any of these I need to know if different Gaussian
>factorizations of an integer n will always have the same sogpf() value.
> sogpf(n) is defined as sum(g_i*e_i) where n has the Gaussian
>factorization n=product(g_i^e_i).
>For example, 2 = (I+1)*(I-1) = (-I)*(1+I)^2 and each factorization
>yields sogpf(2) = 2+I.  Does this hold for all positive integers?

Your very example is, actually, a counterexample:

 2 = (1+I)·(1-I) = (-I)·(1+I)^2

sogpf( (1+I)·(1-I) ) = 1+I + 1 - I = 2 <> 2+I = -I + 2·(1+I) = sogpf( (-I)·(1+I)^2 )

BTW, note that since n = 1·n = 1·...·1·n, it should be sogpf(n) = 1+sogpf(n) = 
1+...+sogpf(n) ! (unless you say that 1 has exponent 0, but we can apply similar tricks 
with associated factors, like (-1)·(-1) and -I·I ).

Regards. Jose Brox