Is this a consecutive-prime/constant-exponent number? A while ago, I had a look at the prime factorization of 27000:

27000 = 23 × 33 × 53

There are two special things about that:

consecutive-prime: The primes are consecutive: 2 is the 1st prime, 3 is the 2nd prime, 5 is the 3rd prime.
constant-exponent: The exponent is the same for every prime (always 3)

Mathematically expressed:

An integer x is a consecutive-prime/constant-exponent number if there exist strictly positive integers n, k, m such that x = pnm × pn+1m × ... × pn+km, where pj is the j-th prime

Your task is to test if a positive integer fulfills these conditions.
Input:
A positive integer &gt; 1, in any reasonable form.
Output:
One of two values, at least one of which has to be constant, indicating whether the input is a consecutive-prime/constant-exponent number.
Edge cases:

primes are truthy, as the factorization for prime p is p1
other numbers that can be written as pm where p is a prime are also truthy.

Rules:

Standard loopholes apply.
No worries about integer overflow, but numbers up to 255 must work.
Shortest code in bytes wins.

Test cases:
Truthy:
2
3
4
5
6
7
8
9
11
13
15
27000
456533

Falsy:
10
12
14
72
10000000

Here is a python script generating some test cases.
The fact that I accepted an answer does not mean that the challenge is over; the winner can still change!
05AB1E, 4 bytes

Ó0ÛË


Try it online!

Explanation

Ó     # get a list of prime exponents
 0Û   # remove leading zeroes
   Ë  # all remaining elements are equal