[seqfan] Re: A Gaussian-integer analog of the sum-of-divisors function
David Corneth
davidacorneth at gmail.com
Wed Apr 27 17:20:31 CEST 2022
I find terms by looking at prime factors of n with multiplicity.
Assuming primes not of the form p = 4*k + 3 can be written in a unique way
as t^2 + u^2 where t and u are coprime and t <= u we can write a(1) = 1,
a(p^e) = ((u + t*i)^(e + 1) - 1) / (u + t*i - 1) * ((u - t*i)^(e + 1) - 1)
/ (u - t*i - 1) and use the mult-property to find the other a(n). Right?
I now have A353151 in draft where I put more terms and b-file using this
idea.
On Wed, Apr 27, 2022 at 2:22 PM Neil Fernandez <primeness at borve.org> wrote:
> 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
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>
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