[seqfan] Re: A Gaussian-integer analog of the sum-of-divisors function
Neil Fernandez
primeness at borve.org
Wed Apr 27 14:22:01 CEST 2022
Hi,
If we allow only one divisor from {x+xi,x-xi} then we will get sums that
are not real, e.g. for a(2).
Neil
In message <CABxCbJ1=Fv1dn=azG3AJYecpQhSngtOijC7C2=Y9=PxRVmDOWQ at mail.gma
il.com>, M. F. Hasler <oeis at hasler.fr> writes
> On Wed, Apr 27, 2022, 07:30 Neil Fernandez <primeness at borve.org>
> wrote:
>> Hi Allan,
>
>> 'm not sure whether I'm applying your rules correctly, but I get
>> values for a(8) and a(10):
>
>
>
>> in first octant or eighth (allowing both boundaries, not
>> 8 itself):
>> 1+i, 2, 2+2i, 4, 4+4i, 8, 1-i, 2-2i, 4-4i};
>
>
> Maybe I get something wrong, but the last three are related to
> earlier ones by a unit ( -i ) and so I would have thought they
> shouldn't be counted a second time.
>
> - M.
>
>> Allan Wechsler <acwacw at gmail.com> writes
>
>> >I have a draft at A353151 for a sequence that is intended to be
>> an analog
>> >of A000205, the sum of the divisors of n.
>> >
>> >This endeavor is a little bit fraught because every Gaussian
>> divisor of n
>> >is one of a set of four "associate" divisors, which are related
>> by a
>> >factors of a Gaussian unit. When we add up the divisors, we
>> only want one
>> >of each associated set; which one shall we choose?
>> >
>> >My choice was to add up the divisors that are the products of
>> powers of
>> >"positivish" Gaussian primes. "Positivish" means that the real
>> part is
>> >positive, and the imaginary part doesn't exceed the real part.
--
Neil Fernandez
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