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

[Joseph Biberstine]

	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?

Joseph Biberstine wrote:
> Here's some numbers for the (a?, since I don't follow your principality
> suggestion) Gaussian analogue.  We are only interested in repetition (ie
> A1414 not A8472).
> 
> Factored over the Gaussian integers, n=Product(g_i^e_i).  Define sogpfr
> (Sum Of Gaussian Prime Factors with Repetition) as sogpfr[n] =
> Sum(g_i^e_i).  This yields a family of nice sequences, all of whose
> obvious integer maps are absent from OEIS.  The following begin with n=2.
> 
> Re[sogpfr[n]] =
> {2, 3, 3, 3, 5, 7, 6, 6, 4, 11, 6, 5, 9, 6, 8, 5, 8, 19, 7, 10, 13, 23,
> 9, 5, 6, 9, 10, 7, 7, 31, 10, 14, 6, 10, 9, 7, 21, 8, 9, 9, 12, ...}
> 
> Im[sogpfr[n]] =
> {1, 0, 4, 2, 1, 0, 7, 0, 5, 0, 4, 4, 1, 2, 8, 4, 1, 0, 8, 0, 1, 0, 7, 6,
> 7, 0, 4, 6, 5, 0, 9, 0, 7, 2, 4, 6, 1, 4, 9, 8, 1, 0, 4, ...}
> 
> Integer part (c of c*sqrt(d)) of Abs[sogpfr[n]] =
> {1, 3, 5, 1, 1, 7, 1, 6, 1, 11, 2, 1, ...}
> 
> Indices of ones in prev seq (or n with Abs[sogpfr[n]]^2 squarefree) =
> {2, 5, 6, 8, 10, 13, 14, 17, 18, 20, 22, 24, 25, 26, 29, 30, 32, 34, 36,
> 37, 38, 41, 42, 45, ...} (compare A81083)
> 
> Squarefree part (d of c*sqrt(d)) of Abs[sogpfr[n]] =
> {5, 1, 1, 13, 26, 1, 85, 1, 41, 1, 13, ...}
> 
> Indices of ones in prev seq (or n with integer Abs[sogpfr[n]]) =
> {3, 4, 7, 9, 11, 19, 21, 23, 27, 31, 33, 43, 47, 49, ...} (compare A4614
> and see Frank's comment)
> 
> Floor[Abs[sogpfr[n]]] =
> {2, 3, 5, 3, 5, 7, 9, 6, 6, 11, 7, 6, 9, 6, 11, 6, 8, 19, 10, 10, 13,
> 23, 11, 7, 9, 9, 10, 9, 8, 31, 13, 14, 9, 10, 9, 9, 21, 8, 12, 12, ...}
> 
> 
> For Mathematica users:
> sogpfr[n_] := Total[Times @@@ FactorInteger[n, GaussianIntegers->True]];
> 
> Barring outed errors I'll probably submit all or most of these sometime
> during the next week.
> 
> -JRB
> 
> Marc LeBrun wrote:
>> 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!)
>>
>>