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 $gcd(a_n,b_n)=1$.

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 $M(137)$.