A problem similar than that of the amicable pairs using the function $\operatorname{rad}(k)$: a first statement or conjecture

Question

In this post we denote the product of distinct primes dividing an integer $k> 1$ as $\operatorname{rad}(k)$, with the definition $\operatorname{rad}(1)=1$, that is the so-called radical of an integer, see this Wikipedia.

A well-known unsolved problem in number theory is related about the so-called amicable pairs. In this post I wanted to do a variation of this old problem and I wondered what should be a first statement or a reasonable conjecture about my sequence.

Definition. I define a the sequence of integers $n\geq 1$, and their correspondings integers $m\geq 1$ to be pairs of $\operatorname{rad}$-amicable numbers if these satisfy (both) the equations $$\sigma(n)=n+\operatorname{rad}(m)$$ and $$\sigma(m)=m+\operatorname{rad}(n),$$ where $\sigma(k)=\sum_{d\mid k}d$ denote the sum of divisors function.

Example. One has that $(33,45)$ is the second $\operatorname{rad}$-amicable pair, satisying $$1+3+11+33=\sigma(33)=33+\operatorname{rad}(45)=33+3\cdot 5.$$

Question. I would like to know what can be a first statement about previous defined sequence of $\operatorname{rad}$-amicable numbers, having a good mathematical content or well a conjecture about this sequence of integers $n\geq 1$ or pairs $(n,m)$. Many thanks.

Thus we evoke the scenario that this sequence is interesting and we need to explore with mathematics a first statement or conjecture related to our sequence. What should be a first step in the study of our sequence?

Computational fact. Our sequence of integers $n$ starts as $$6,33,45,46,52,1799,6625,\ldots$$ being thus the first few pairs $(n,m)$ these $(6,6),(33,45),(45,33),(46,52),(52,46),\ldots$

Answer 1

The following PARI/GP program searches efficiently for pairs doing the job but it is not guaranteed that all pairs are found.

The idea is : For every $s$ create all $u$ such that the radical of $u$ is $\sigma(s)-s$ and just verify whether this $u$ satisfies the other condition.

The program allows all exponents in $u$ to be upto $5$ (this can be changed). Upto $s=10^6$ , the following pairs were found :

? for(s=2,10^6,if(issquarefree(sigma(s)-s)==1,t=sigma(s)-s;x=factor(t);x=compone
nt(x,1);forvec(z=vector(length(x),j,[1,5]),u=prod(j=1,length(z),x[j]^z[j]);if(u<
=s,if(sigma(u)-u==rad(s),print([u,s]))))))
[6, 6]
[33, 45]
[46, 52]
[1799, 6625]
[7905, 17757]
[142310, 168730]
[40067, 414613]
?

All those pairs (and many additional pairs) occur upto $2\cdot 10^7$ with limit exponent $2$ :

? for(s=2,2*10^7,if(issquarefree(sigma(s)-s)==1,t=sigma(s)-s;x=factor(t);x=compo
nent(x,1);forvec(z=vector(length(x),j,[1,2]),u=prod(j=1,length(z),x[j]^z[j]);if(
u<=s,if(sigma(u)-u==rad(s),print([u,s]))))))
[6, 6]
[33, 45]
[46, 52]
[1799, 6625]
[7905, 17757]
[142310, 168730]
[40067, 414613]
[1077890, 1099390]
[1156870, 1292570]
[1511930, 1598470]
[1669910, 2062570]
[2236570, 2429030]
[2728726, 3077354]
[4246130, 4488910]
[5123090, 5504110]
[5385310, 5812130]
[4532710, 6135962]
[6993610, 7158710]
[7288930, 8221598]
[8619765, 9627915]
[8826070, 10043690]
[10254970, 10273670]
[8754130, 10893230]
[9478910, 11049730]
[1200277, 11785787]
[17257695, 17578785]
[14426230, 18087818]
[17041010, 19150222]
?