[seqfan] Re: A Gaussian-integer analog of the sum-of-divisors function

[Neil Fernandez]

Hi again Allan,

For sequence a(n) which begins

1, 5, 4, 13, 10, 20, 8, 29, 13, 48, 12, 52, 20, 40, 40, 61, 26, 65, 20,
124, 32, 60, 24, 116, 63, 98, 40, 104, 40, 192, 32, 125, 48, 124, 80,
169, 50, 100, 80, 276, 52, 160, 44, 156, 130, 120, 48, 244, 57, 301,
104, 254, 68, 200, 120, 232, 80, 194, 60, 496, 74, 160, 104, 253, 206,
240, 68, 320, 96, 384, 72, 377, 90, 240, 252, 260, 96, 392, 80, 580,
121, 258, 84, 416, 260, 220, 160, 348, 106, 624, 160, 312, 128, 240,
200, 500, 116, 285, 156, ...

we can also define the following sequences:

* numbers for which a(n) is prime:

2, 4, 8, 9, 16, 128, 256, 729, 841, 1024, 1369, 2401, 2738, 2809, 3481,
4096, 5041, 5618, ...

* numbers for which a(n) is a Gaussian prime:

2738, 5618, 13456, 24649, 31250, 32761, 37538, 40804, 43808, 75076,
77618, 97969, ...

* numbers for which a(2^n) is prime

1,2,3,4,7,8,10,12,18,20,22, ...

Neil


In message <CADy-sGEbT1=HC57LbMtRQGqdz+--KyrQ3ysbesS_+jdz3oHsgA at mail.gma
il.com>, 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.
>
>By specifying a canonical set of Gaussian primes, we ensure
>number-theoretic multiplicativity over the Gaussian integers; and it very
>prettily turns out that this implies ordinary integer multiplicativity, so
>if m and n are relatively prime, a(mn) = a(m)a(n).
>
>My data so far is:
>
>1, 5, 4, 13, 10, 20, 8, 25, 13, 50, 12, 52, 20, 40, 40...
>
>I would love confirmation, and more data. Note that 1, 5, and 10 are
>analogous to multiperfect numbers, of orders 1, 2, and 5 respectively.


-- 
Neil Fernandez