Problem 541

Divisibility of Harmonic Number Denominators

The n^thharmonic number H_n is defined as the sum of the multiplicative inverses of the first n positive integers, and can be written as a reduced fraction a_n/b_n.
$H_{n} = \sum_{k = 1}^{n} \frac{1}{k} = \frac{a_{n}}{b_{n}}$ , with $gcd (a_{n}, b_{n}) = 1$ .

Let M(p) be the largest value of n such that b_n 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H_n被定义为前n个正整数的倒数和，可以写成最简分数a_n/b_n的形式。
$H_{n} = \sum_{k = 1}^{n} \frac{1}{k} = \frac{a_{n}}{b_{n}}$ ，其中 $gcd (a_{n}, b_{n}) = 1$ 。

令M(p)为使得b_n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)。