Sequence of Irrational Numbers

[lemniscatus]

Let

x = f(n)
y = g(n)
z = h(x,y) = x^(1/y)

where

x,y are rational numbers
n is a natural number
z is a real number
x > 0

Problem:

Let p,q be natural numbers.

If z is a natural number for n = p and
z is an irrational number for all n = p - q > 1 and q > 1,
can we have a natural number z when q = 1?

Note:

f and g are functions that must contain at least
one operation on (or involving) n.

Example:

x = 2n + (-1)^n
y = 2n - 17
z = x ^ (1/y)

n f g z
====================================
9 17 1 17
8 17 -1 0.0588235294
7 13 -3 0.4252903703
6 13 -5 0.5987028555
5 9 -7 0.7305999556
4 9 -9 0.7833810369
3 5 -11 0.8638876637
2 5 -13 0.8835539578
1 1 -15 1

Here z is a natural number for n = p = 9
and irrational for n = p - q > 1, q > 1.

Obviously, z is not a natural number for
n = p - 1 in this example.