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The Millionth Number with at Least One Million Prime Factors

 Published on Sunday, 3rd December 2017, 10:00 am and solved by 827
Difficulty: Level 25 [63%]

Problem 615

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 $123454321$.



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