[seqfan] A Gaussian-integer analog of the sum-of-divisors function
Allan Wechsler
acwacw at gmail.com
Wed Apr 27 03:40:01 CEST 2022
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.
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