Eccoci.

Any positive integer can be written as a product of prime powers: $p_1^{a_1} \times p_2^{a_2} \times \cdots \times p_k^{a_k}$,
where $p_i$ are distinct prime integers, $a_i \gt 0$ and $p_i \lt p_j$ if $i \lt j$.

A decreasing prime power positive integer is one for which $a_i \ge a_j$ if $i \lt j$.
For example, $1$, $2$, $15=3 \times 5$, $360=2^3 \times 3^2 \times 5$ and $1000=2^3 \times 5^3$ are decreasing prime power integers.

Let $C(n)$ be the count of decreasing prime power positive integers not exceeding $n$.
$C(100) = 94$ since all positive integers not exceeding $100$ have decreasing prime powers except $18$, $50$, $54$, $75$, $90$ and $98$.
You are given $C(10^6) = 922052$.

Find $C(10^{13})$.