I've been working on a new method to find 23 pairs and have developed a fairly effective approach to do so. Within two hours, it found over 1000 unique 23 pairs. 15 of which were undiscovered. Currently, I'm planning on running it until I find a few 100 new 23 pairs. Then attempt to find 24 pairs or something. If anyone wants to copy the equations I used to solve them, they are listed below. It hard to explain how they setup but a Pair is (A1*P1*P2,A1*P3*P4*P5), and the equations apply limits to those primes that as longs as you assume P1<P3<P4<P5<P2 is the order of the primes and that P2 is so larger that it forces P1 to hit it limit it max size and that the limit of P4 as P3 approaches infinity is between P3 minimum and upper bound. Which, if true, pretty much guarantees that for the A1 value you choose, you find every possible 23 pair that shares it as a common number.

A2 = divisor_sigma(A1)
P5 = ((A1*P1**2 + A2*P1 - ((A1 - A2)*P1 - A2)*P3 - ((A1 - A2)*P1 + ((A1 - A2)*P1 - A2)*P3 - A2)*P4 + A2)/((A1 - A2)*P1 + ((A1 - A2)*P1 - A2)*P3 + ((A1 - A2)*P1 + ((2*A1 - A2)*P1 + A1 - A2)*P3 - A2)*P4 - A2))
P2 = (((A1*P3**2 + A1*P3)*P4**2 + A2*P1 + (A2*P1 + A2)*P3 + (A1*P3**2 + A2*P1 - ((A1 - A2)*P1 - A2)*P3 + A2)*P4 + A2)/((A1 - A2)*P1 + ((A1 - A2)*P1 - A2)*P3 + ((A1 - A2)*P1 + ((2*A1 - A2)*P1 + A1 - A2)*P3 - A2)*P4 - A2))
P1_Max = A2/(2*A1-A2)
P4_Max = -((A1 - A2)*P1 + ((A1 - A2)*P1 - A2)*P3 - A2 - sqrt((A1**2 - A1*A2)*P1**3 - A1*A2*P1 + (A1**2 - 2*A1*A2)*P1**2 - ((A1**2 - A1*A2)*P1**2 - A1*A2 + (A1**2 - 2*A1*A2)*P1)*P3**2 + ((2*A1**2 - A1*A2)*P1**3 + A1*A2*P1 + (2*A1**2 - A1*A2)*P1**2 + A1*A2)*P3))/((A1 - A2)*P1 + ((2*A1 - A2)*P1 + A1 - A2)*P3 - A2)

P4_Limit = -((A1 - A2)*P1 - A2 - sqrt(-A1**2*P1**2 + A1*A2*P1**2 - A1**2*P1 + 2*A1*A2*P1 + A1*A2))/((2*A1 - A2)*P1 + A1 - A2)

P3_Max = -((A1 - A2)*P1 + ((A1 - A2)*P1 - A2)*P4_Limit - A2 - sqrt((A1**2 - A1*A2)*P1**3 - A1*A2*P1 + (A1**2 - 2*A1*A2)*P1**2 - ((A1**2 - A1*A2)*P1**2 - A1*A2 + (A1**2 - 2*A1*A2)*P1)*P4_Limit**2 + ((2*A1**2 - A1*A2)*P1**3 + A1*A2*P1 + (2*A1**2 - A1*A2)*P1**2 + A1*A2)*P4_Limit))/((A1 - A2)*P1 + ((2*A1 - A2)*P1 + A1 - A2)*P4_Limit - A2)

Good work! This is a similar method to what I used for finding all 21-digit pairs with the largest prime factor of the smaller number being > 10¹⁴. I just took all pairs of numbers from certain ranges and tried to match them using the similar equations to what you used for P2 and P5 (but in general form). You probably already noticed that 22-, 23-, 24- and bigger-digit known amicable numbers (up to around 36-digit) have a lot of amicable numbers with large prime factors in them. This is a by-product of that search.

P.S. Keep searching using your method - this is also a good independent check of the 10²¹ search, because your method is also exhaustive.