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Problem 615


Problem 615


The millionth number with at least one million prime factors

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⋅2⋅2⋅2⋅2
  • 48=2⋅2⋅2⋅2⋅3
  • 64=2⋅2⋅2⋅2⋅2⋅2
  • 72=2⋅2⋅2⋅3⋅3
  • 80=2⋅2⋅2⋅2⋅5
  • 96=2⋅2⋅2⋅2⋅2⋅3

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.


第一百万个拥有至少一百万个质因数的数

考虑所有拥有至少5个质因数的自然数,这些质因数不必完全不同。
将这些数按从小到大排序,所构成的列表开头是这几个数:

  • 32=2⋅2⋅2⋅2⋅2
  • 48=2⋅2⋅2⋅2⋅3
  • 64=2⋅2⋅2⋅2⋅2⋅2
  • 72=2⋅2⋅2⋅3⋅3
  • 80=2⋅2⋅2⋅2⋅5
  • 96=2⋅2⋅2⋅2⋅2⋅3

所以,第五个拥有至少5个质因数的数是80。

找出第一百万个拥有至少一百万个质因数的数。
将你的答案对123454321取余。