174 on consecutive-prime-constant-exponent:
# N = smallest root of N
^(?=(x+?)(((\1(x+))(?=(\4*)\1*$)\4*(?=\5$\6))*)x$)\2
# Assert that there exists no trio of prime numbers such that N is divisible by the
# smallest and largest prime but not the middle prime.
(?!(((x+)(?=\9+$)(x+))(?!\8+$)(x+))\7*(?=\7$)(?!(\11\10?)?((xx+)\14+|x?)$))
# Assert that N is square-free (its prime factors all have single multiplicity)
((?=(xx+?)\16*$)(?=(x+)(\17+$))\18(?!\16*$))*x$
The overall algo is mine, but all I did was copy three of your regexes, change a + to a * in perfect powers, and adjust the back…