## Integers with decreasing prime powers

### Problem 578

Any positive integer can be written as a product of prime powers: `p`_{1}^{a1} × `p`_{2}^{a2} × ... × `p _{k}^{ak}`,

where

`p`are distinct prime integers,

_{i}`a`> 0 and

_{i}`p`<

_{i}`p`if

_{j}`i`<

`j`.

A *decreasing prime power* positive integer is one for which `a _{i}` ≥

`a`if

_{j}`i`<

`j`.

For example, 1, 2, 15=3×5, 360=2

^{3}×3

^{2}×5 and 1000=2

^{3}×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}).