[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


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).


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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