Is there a number $\mathscr{D}_2 \neq \mathscr{D} = {{3003}^2}\cdot{22021}$ satisfying a certain condition? (Note:  This question is tangentially related to this earlier one.)

Let $$\sigma(x) = \sum_{d \mid x}{d}$$ denote the sum of divisors of $x \in \mathbb{N}$, where $\mathbb{N}$ is the set of natural numbers or positive integers.

Recall that a Descartes number is an odd number $n = km$, with $1 &lt; k$, $1 &lt; m$, satisfying $$\sigma(k)(m+1)=2km.$$  ($m$ is called the quasi-Euler prime of $n$.)  Note that we define $\sigma(m) := m + 1$ even when $m$ is composite (that is, we pretend that $m$ is prime).

Notice that the lone Descartes number $\mathscr{D}$ that is known is
$$\mathscr{D} = k'm' = {{3003}^2}\cdot{22021}.$$


  In particular, note that:
  $$\frac{\sigma(k')}{m'} + \frac{\sigma(m')}{k'} = \frac{670763}{819} \approx 819.002,$$
  where 
  $$\lfloor{\frac{670763}{819}}\rfloor = 819 = 2\cdot{{3003}^2} - \sigma({{3003}^2}) = D({{3003}^2}),$$
  and $D(x)=2x-\sigma(x)$ is the deficiency of the positive integer $x$.


Here are my questions:


  (1) Is there a number $k_2 \times m_2 = \mathscr{D}_2 \neq \mathscr{D} = {{3003}^2}\cdot{22021}$ satisfying
  $$\lfloor{\frac{\sigma(k_2)}{m_2} + \frac{\sigma(m_2)}{k_2}}\rfloor = D(k_2)?$$


Note that I have no restrictions on $k_2$, $m_2$, and therefore, $\mathscr{D}_2$.


  (2) When does the relationship
  $$\lfloor{\frac{\sigma(k_3)}{m_3} + \frac{\sigma(m_3)}{k_3}}\rfloor = D(k_3)$$
  hold, for some integers $1 &lt; k_3$, $1 &lt; m_3$?
After trying to search (in Sage Cell Server) for examples satisfying the equation
$$\lfloor{\frac{\sigma(k_3)}{m_3} + \frac{\sigma(m_3)}{k_3}}\rfloor = D(k_3)$$
for some integers $1 \leq k_3 \leq 1000$, $1 \leq m_3 \leq 1000$, I believe that it holds for most $k_3$ and $m_3$:

Pari/GP script #1

for(x=1, 1000, for(y=1, 1000, if(floor((sigma(x)/y)+(sigma(y)/x))==2*x-sigma(x),print(x,"   ",y))))


Output #1

I have suppressed the output - there are several lines returned.


Pari/GP script #2

for(x=1, 1000, for(y=1, 1000, if(floor((sigma(x)/y)+(sigma(y)/x))==2*x-sigma(x) &amp;&amp; gcd(x,y)==1,print(x,"   ",y))))


Output #2

I have suppressed the output - there are several lines returned.


Pari/GP script #3

for(x=1, 1000, for(y=1, 1000, if(floor((sigma(x)/y)+(sigma(y)/x))==2*x-sigma(x) &amp;&amp; gcd(x,y)==1 &amp;&amp; issquare(x),print(x,"   ",y))))


Output #3

9   20
9   26
9   35
9   43
9   47
25   168
25   248
25   266
25   268
25   297
25   314
25   326
25   333
25   413
25   427
25   437
25   479
25   487
25   491
49   660
49   768
49   852
49   858
49   940
49   992
484   25
After trying to search (in Sage Cell Server) for examples satisfying the equation
$$\lfloor{\frac{\sigma(k_3)}{m_3} + \frac{\sigma(m_3)}{k_3}}\rfloor = D(k_3)$$
for some integers $1 \leq k_3 \leq 1000$, $1 \leq m_3 \leq 1000$, I believe that it holds for most $k_3$ and $m_3$:

Pari/GP script #1

for(x=1, 1000, for(y=1, 1000, if(floor((sigma(x)/y)+(sigma(y)/x))==2*x-sigma(x),print(x,"   ",y))))


Output #1

I have suppressed the output - there are several lines returned.


Pari/GP script #2

for(x=1, 1000, for(y=1, 1000, if(floor((sigma(x)/y)+(sigma(y)/x))==2*x-sigma(x) &amp;&amp; gcd(x,y)==1,print(x,"   ",y))))


Output #2

I have suppressed the output - there are several lines returned.


Pari/GP script #3

for(x=1, 1000, for(y=1, 1000, if(floor((sigma(x)/y)+(sigma(y)/x))==2*x-sigma(x) &amp;&amp; gcd(x,y)==1 &amp;&amp; issquare(x),print(x,"   ",y))))


Output #3

9   20
9   26
9   35
9   43
9   47
25   168
25   248
25   266
25   268
25   297
25   314
25   326
25   333
25   413
25   427
25   437
25   479
25   487
25   491
49   660
49   768
49   852
49   858
49   940
49   992
484   25