$x$ power floor $x$ ?

If x = 1, then $\alpha$ = 1

If 2 $\leq$ x < 3, the exponent will be 2, so:

• x = $\sqrt 4$ = 2 $\rightarrow$ $\alpha$ = 4

• x = $\sqrt 5$ $\rightarrow$ $\alpha$ = 5

• x = $\sqrt 6$ $\rightarrow$ $\alpha$ = 6

• x = $\sqrt 7$ $\rightarrow$ $\alpha$ = 7

• x = $\sqrt 8$ $\rightarrow$ $\alpha$ = 8

We stop at $\sqrt 8$ because $\lfloor$ $\sqrt 9$ $\rfloor$ = 3, which changes the exponent. So:

If 3 $\leq$ x < 4, the exponent will be 3, so:

• x = $\sqrt[3]27$ = 3 $\rightarrow$ $\alpha$ = 27

• x = $\sqrt[3]28$ $\rightarrow$ $\alpha$ = 28

$\vdots$

• x = $\sqrt[3]63$ $\rightarrow$ $\alpha$ = 63

Again, for x = $\sqrt[3]64$ the exponent changes, so:

If 4 $\leq$ x < 5, the exponent will be 4, so:

• x = $\sqrt[4]256$ = 4 $\rightarrow$ $\alpha$ = 256

• x = $\sqrt[4]257$ $\rightarrow$ $\alpha$ = 257

$\vdots$

• x = $\sqrt[4]624$ $\rightarrow$ $\alpha$ = 624

Again, for x = $\sqrt[4]625$ the exponent changes, so:

If 5 $\leq$ x < 6, the exponent will be 5. But $5^5$ = 3125 > 1000.

So, $\alpha$ can only be {1, 4, 5, 6, 7, 8, 27, 28, 29, ... ,63, 256, 257, 258,.... 624}, which gives 412 possible values