Problem Archive

Consider the natural numbers having at least $5$ prime factors, which don't have to be distinct.
Sorting these numbers by size gives a list which starts with:

• $32=2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$
• $48=2 \cdot 2 \cdot 2 \cdot 2 \cdot 3$
• $64=2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$
• $72=2 \cdot 2 \cdot 2 \cdot 3 \cdot 3$
• $80=2 \cdot 2 \cdot 2 \cdot 2 \cdot 5$
• $96=2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3$
• $\cdots$

So, for example, the fifth number with at least $5$ prime factors is $80$.

Find the millionth number with at least one million prime factors.
Give your answer modulo $123454321$.