# Does there exist an odd natural number $N = xy$ satisfying $D(x)D(y) = 2s(x)s(y)$?

Preamble: I apologize in advance if what I am asking for in this question, I could get an answer easily so myself, for example by coding a short Mathematica script. It is just that I have not yet studied the rudiments of Mathematica (or even Python) scripting, so I am not there yet.

This question is a follow-up to this earlier MSE question.

Let $\sigma(z)$ be the sum of the divisors of $z \in \mathbb{N}$. Denote the deficiency of $z$ by $D(z) := 2z - \sigma(z)$ and the sum of the aliquot parts of $z$ by $s(z) := \sigma(z) - z$.

Here is my question:

Does there exist an odd natural number $N = xy$ satisfying $D(x)D(y) = 2s(x)s(y)$?

• Just to add a code. Same instructions than before (my English was bad): search in Internet Sage Cell Server, and copy and paste with the mouse the following code (is a line) for (x = 1, 1000,for (y = 1, 1000,if(x%2==1&&y%2==1&&(2*x-sigma(x))*(2*y-sigma(y))==2*(sigma(x)-x)*(sigma(y)-y),print (x*y)))) Secondly choose GP as Language and press Evaluate. Then when the green box of the output will black, means that the program has finished (in this example there isn't output for the range $1\leq x,y\leq 1000$, where the syntax variable%2==1 means that the variable is an odd integer).
– user243301
Commented Apr 23, 2018 at 18:01
• Thank you very much for this, @user243301! This will be a big help for me when I test my number-theoretic conjectures. ^_^ Commented Apr 24, 2018 at 1:40

The answer is "YES" for the Descartes spoof $$\mathscr{D} = 198585576189 = {22021}\cdot{3003}^2,$$ where we set $x = 22021$ and $y = {3003}^2$. (That is, if we pretend that $x = 22021$ is prime.)
We "obtain" $$D(x)D(y) = 18034380 = 2s(x)s(y).$$