Integers with decreasing prime powers (noch nicht übersetzt)

Any positive integer can be written as a product of prime powers: p₁^a₁ × p₂^a₂ × ... × 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 ≥ a_j if i < 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).