## Divisibility of Harmonic Number Denominators

### Problem 541

The `n`^{th} **harmonic 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}$H_n = \displaystyle \sum_{k=1}^n \frac 1 k = \frac {a_n} {b_n}$, with $\text {gcd}(a_n, b_n)=1$.

Let `M`(`p`) be the largest value of `n` such that `b _{n}` is not divisible by

`p`.

For example, `M`(3) = 68 because $H_{68} = \frac {a_{68}} {b_{68}} = \frac {14094018321907827923954201611} {2933773379069966367528193600}$, `b`_{68}=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).