Question related to amicable numbers not divisible by 3 In a paper by W. Borho and S. Battiato, the authors show that there do exist odd amicable numbers which are not divisible by 3. For their examples they found a common factor:
$a=5^{4}\cdot 7^{3}\cdot11^{2}\cdot 13^{2} \cdot 17^{2} \cdot 19 \cdot 61^{2} \cdot 97 \cdot 307$. 
What I am trying to understand is if every odd amicable pair that is not divisible by $3$ (either member) must have this common factor $a$ or if this was just one example of such an $a$?
Link to paper provided below in comment, sorry for poor link
 A: The paper presented was a counterexample of a conjecture on amicable numbers. The belief was that every pair of odd amicable numbers $(x,y)$ followed the rule $3 \mid x$ and $3 \mid y$. They used the common factor $a= 5^4⋅7^3⋅11^2⋅13^2⋅17^2⋅19⋅61^2⋅97⋅307$ to construct their counterexamples. However, this common factor need not divide all odd amicable pairs. The authors meant that $a$ divided all of their pairs used to disprove the conjecture.
The above can be seen clearly by considering two unsolved problems in mathematics:
$(1)$ Are there any pairs of amicable numbers with one odd and one even positive integer?
$(2)$ Are there any pairs of relatively prime amicable numbers?
If we knew that $a$ divided all pairs of amicable numbers where $3$ didn't, then we would know that all pairs of odd amicable numbers cannot be relatively prime. The only possibility for the same is if there exist an odd - even pair of amicable numbers, which are relatively prime. Then, we would get the following relation:
$$(1)=\mathbf{False} \implies (2)=\mathbf{False}$$
$$(2)=\mathbf{True} \implies (1)=\mathbf{True}$$
The paper by W. Borho and S. Battiato was written in the year $\mathrm1988$. Here is a line from the paper - "On Relatively Prime Amicable Numbers" by Paul Pollack, written in the year $\mathrm2000$ -

In fact, if $N$ and $M$ are assumed to have opposite parity, then
  $10^{67}$ can be replaced with $10^{121}$.

Here, the author is talking about the lower bounds for relatively prime pairs of amicable numbers $(N,M)$. This line proves that we were unaware of a relation such as the above, between the two conjectures. This is only possible if the authors of the $\mathrm1988$ paper meant that $a$ divided all of their pairs only.
It need not be that $a$ does not divide all pairs which are not divisible by $3$, but this would be a highly unlikely scenario.
Citations:
Paper by Paul Pollock : On Relatively Prime Amicable Pairs
Paper by W. Borho and S. Battiato : Are There Odd Amicable Numbers Not Divisible by Three?
