What is the relationship between number of perfect squares and number of perfect cubes?

Question

If S is the "number" of perfect squares between 1 and a specific integer N and Q is the "number" of perfect cubes between 1 to N. How we can show the relationship between S and Q?

Please explain with mathematical proof.

Answer 1

You have

$$S = \lfloor \sqrt{N} \rfloor$$

Indeed, the square between 1 and N are :

$$1, 2^2, 3^2, \cdots, (S-1)^2, S^2$$

With $S$ an integer such that $\sqrt{N}-1 < S \leq \sqrt{N}$

Same idea with the cube :

$$Q = \lfloor \sqrt[3]{N} \rfloor$$

Now, I'm not sure there is an explicit relation between the two. On average, $\frac{S}{Q} \sim N^{\frac{1}{6}}$