haha, I forgot to simplify it

Without using re, this is probably the most efficient implementation

Very clever, clearly understand the number math of power of 3

An complicated implementation without the understanding of the underline number math

What about the case: word1 = 'ab1cd', word2 = 'abcd'? I think this should be True, but your solution would return False.

I see. Thanks!

This may not be a correct implementation. For example: A = [3, 2, 3, 4, 5], a = 3, b = 4.
Using your function would return False, because A.index(3) = 0, A.index(4) = 3

Probably the most elegant implementation of this problem.

Sorry, this is really a stupid implementation. Early in the morning, my brain is not working

Lots of errors in your test cases!
13 is not a Mersenne prime as 14 is not a power of 2
Neither are 17, 19, ...
I think I have the correct implementation, please use it to generate real test cases.

I think this is probably the most efficient implementation

My factors() could be optimized by add d and N//d at the same time when N%d==0 and do it only to int(N**0.5)+1

Maybe I'm not smart enough, I don't understand how one can do this with just one pass of the string.

Since the initital 5 test cases has a range 2 to 1,000,000, which would generate all legit numbers for all other testcases. So this is probably the most efficient implementation.