Ok for quite some time i did indeed write alot of analysis code for integer sequences, and the most important thing for me was to not get caught up in the question "is this a random pattern or not?" Because it turns out that even if it is indeed completely random, there will still be underlying principles of number theory that will allow us to have at least some kind of expectation.
But in a practically applicable sense to what the human brain is capable of doing, a high level of complexity in terms of the number theoretical meaning of the arithmetic functions that produce a given sequence are as good as random to the person unaware of their exact nature.
Always start with the basics is my principle.
Also please try to present you pattern in a way that is their most simplied form. In your example it is only the exponents of the numerator and denominator that exhibit variance, so the better what to introduce the question is to explain that, and ask if we can predict the pair of sequences that are not constant:
1415, 1648, 3474, 5984, 6197, 8739
893, 1040, 2192, 3776, 3910, 5514
This makes it far easier for your computer and fair easier for you, the numbers in the original sequence are just unnecessarily large.