Divisibility of Harmonic Number Denominators

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 $M(137)$.

To find $M(137)$, we need to calculate the harmonic numbers and check the divisibility of their denominators by $137$ until we find the largest value of $n$ where the denominator is not divisible by $137$.

We can solve this problem by using Python. Below is the implementation:

“`python
from fractions import Fraction

def gcd(a, b):
# Function to calculate the greatest common divisor
while b:
a, b = b, a % b
return a

def M(p):
n = 1 # Start with the first harmonic number
while True:
# Calculate the denominator of the current harmonic number
denominator = 1
for k in range(1, n+1):
denominator *= k

# Check if the denominator is divisible by p
if denominator % p != 0:
prev_denominator = denominator
else:
return n-1 if prev_denominator is None else n-2

n += 1

# Find M(137)
result = M(137)
print(result)
“`

The code begins by defining a function `gcd` to calculate the greatest common divisor. This will be used later to find the reduced fraction.

The `M` function takes an input `p` and iterates through the harmonic numbers until it finds a denominator that is not divisible by `p`. It initializes `n` to `1` and enters an infinite loop (`while True`).

Inside the loop, the code calculates the denominator of the current harmonic number by multiplying all the positive integers up to `n`. It then checks if the denominator of the current harmonic number is divisible by `p`. If it is not, it stores the previous denominator in a variable `prev_denominator` and continues to the next iteration. If it is divisible by `p`, it returns `n-1` if `prev_denominator` is `None` (indicating that this is the first harmonic number), or `n-2` (indicating the largest value of `n` where the denominator is not divisible by `p`).

Finally, the code calls the `M` function with `137` as the input and prints the result.

Executing this code will provide the value of `M(137)`, which is the largest value of `n` such that the denominator of the harmonic number is not divisible by `137`.

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