Find values of n for which expression is integer

Question

I have an expression for which I want ot find values of n that will make it an integer. Here is the expression:

(n^(3/2) - n)^(1/3)
or
n^(1/2) - n^(1/3)

I want to find all positive values of n for which the above expression will be an Integer. Can I do that with Mathematica?

Thanks.

Answer 1

Using InverseFunction

f := Evaluate@InverseFunction[(#^(3/2) - #)^(1/3) &]

a = N[f /@ Range[3]]

{2.1479, 5.73535, 11.3776}

The above values will produce integers, e.g.

(#^(3/2) - #)^(1/3) & /@ a

{1., 2., 3.}