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# Problem 578

## Integers with decreasing prime powers

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, $a_i > 0$ and $p_i<p_j$ if $i<j$.

A decreasing prime power positive integer is one for which $a_i \ge a_j$ if $i\lt j$.
For example, 1, 2, 15=3×5, 360=23×32×5 and 1000=23×53 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(106) = 922052.

Find C(1013).

## 质因数幂次下降的整数

C(100) = 94，因为除了18，50，54，75，90和98外，其它所有不超过100的正整数都是质因数幂次下降的。