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

Neil Fernandez primeness at borve.org
Wed Apr 27 13:30:18 CEST 2022

```Hi Allan,

I'm not sure whether I'm applying your rules correctly, but I get
different values for a(8) and a(10):

a(8):

divisors in first octant or eighth (allowing both boundaries, not
allowing 8 itself):
{1, 1+i, 2, 2+2i, 4, 4+4i, 8, 1-i, 2-2i, 4-4i};

sum: 29

a(10):

divisors (as above)
{1, 1+i, 2, 2+i, 3+i, 4+2 i, 5, 5+5i, 10, 1-i, 2-i, 3-i, 4-2i, 5-5i};

sum: 48

And for sequence a(n):

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, ...

If we count n as a divisor of n (as in A000203), we get:

2, 7, 7, 17, 15, 26, 15, 37, 22, 58, 23, 64, 33, 54, 55, 77, 43, 83, 39,
144, 53, 82, 47, 140, 88, 124, 67, 132, 69, 222, 63, 157, 81, 158, 115,
205, 87, 138, 119, 316, 93, 202, 87, 200, 175, 166, 95, 292, 106, 351,
155, 306, 121, 254, 175, 288, 137, 252, 119, 556, 135, 222, 167, 317,
271, 306, 135, 388, 165, 454, 143, 449, 163, 314, 327, 336, 173, 470,
159, 660, 202, 340, 167, 500, 345, 306, 247, 436, 195, 714, 251, 404,
221, 334, 295, 596, 213, 383, 255, ...

Neil

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

```