# 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).