I've defined a sequence of sequences $\{x^n\}$ as follows
$x^1=(1^2,2^2,3^2,4^2,5^2,...)$
$x^2=(1,2^2,3^2,4^2,5^2,....)$
$x^3=(1,2,3^2,4^2,5^2,...)$
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and for each $n$ fixed, I am trying to determine $|\{j: x^n_{j} \leq k\}|$. In other words, for a sequence $x^n$, if I look at the term $x^n_{k}$, I want to calculate how many terms of the sequence satisfy $x^n_{j} \leq k$.
Hint: Note that as long as you are in the squares, you have the same number of terms. The term $p^2$ is the $p^{\text{th}}$ term. If you are in the first powers, $q$ is the $q^{\text{th}}$ term. When are you in each regime?