Square Root Everywhere

In his dream, Chris wrote a function in Python, which takes a positive integer as the parameter.

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def func(n):
    if n == 1:
        return 1
    val = 0
    for i in range(int(sqrt(n))):
        val += func(int(sqrt(n)))
    return val

How many distinct inputs $n$ are there such that this function returns 627372?