The prime tower factorization of a number is defined in A182318.
Two consecutive numbers cannot have a common prime factor; however, their prime tower factorizations can share a prime number.
For example, the prime tower factorizations of 8 and 9, that is, 2^3 and 3^2, share the prime numbers 2 and 3.
We can also find triples of consecutive numbers whose prime tower factorizations share a prime number:
 if n is an odd squarefree number > 1, then the prime tower factorizations of n^21, n^2 and n^2+1 share the prime number 2,
 the prime tower factorizations of 5344, 5345 and 5346 share the prime number 5.
Also, the prime tower factorizations of:
 342, 343, 344 and 345 share the prime number 3,
 99125, 99126, 99127, 99128 and 99129 share the prime number 3,
 72470 ... 72480 share the prime number 2,
 1674274 ... 1674288 share the prime number 2.
Are there tuples of more than 15 consecutive numbers with such a property?
