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。