Problem 541
Divisibility of Harmonic Number Denominators
The nthharmonic number Hn is defined as the sum of the multiplicative inverses of the first n positive integers, and can be written as a reduced fraction an/bn.
$H_n = \displaystyle \sum_{k=1}^n \frac{1}{k} = \frac {a_n} {b_n}$, with$\text {gcd}(a_n, b_n)=1$.
Let M(p) be the largest value of n such that bn is not divisible by p.
For example, M(3) = 68 because $H_{68} = \frac{a_{68}}{b_{68}} = \frac{14094018321907827923954201611}{2933773379069966367528193600}$,
b68=2933773379069966367528193600 is not divisible by 3, but all larger harmonic numbers have denominators divisible by 3.
You are given M(7) = 719102.
Find M(137).
和谐数分母的整除性
第n个和谐数Hn被定义为前n个正整数的倒数和，可以写成最简分数an/bn的形式。
$H_n = \displaystyle \sum_{k=1}^n \frac{1}{k} = \frac {a_n} {b_n}$， 其中$\text {gcd}(a_n, b_n)=1$。
令M(p)为使得bn不能被p整除的最大的n。
例如，M(3) = 68，因为$H_{68} = \frac{a_{68}}{b_{68}} = \frac{14094018321907827923954201611}{2933773379069966367528193600}$，b68=2933773379069966367528193600不能被3整除，但是所有更大的和谐数的分母都能被3整除。
已知M(7) = 719102。
求M(137)。