Eight Divisors

The eight divisors of $24$ are $1, 2, 3, 4, 6, 8, 12$ and $24$.
The ten numbers not exceeding $100$ having exactly eight divisors are $24, 30, 40, 42, 54, 56, 66, 70, 78$ and $88$.
Let $f(n)$ be the count of numbers not exceeding $n$ with exactly eight divisors.
You are given $f(100) = 10$, $f(1000) = 180$ and $f(10^6) = 224427$.
Find $f(10^{12})$.

To find the value of $f(10^{12})$, we first need to understand the pattern of numbers that have exactly eight divisors.

The divisors of a number $n$ can be found by iterating from $1$ to $\sqrt{n}$ and checking if the number is divisible by the current iteration. If it is, both the current iteration and the result of dividing $n$ by the current iteration will be divisors of $n$.

Using this approach, we can count the number of divisors for each number up to $n$. If the count is exactly eight, we increment a counter variable.

Now, let’s write Python code to implement this logic and find the value of $f(10^{12})$.

“`python
import math

def count_divisors(n):
count = 0
sqrt_n = int(math.sqrt(n))

for i in range(1, sqrt_n + 1):
if n % i == 0:
count += 1
if (n // i) != i:
count += 1

return count

def f(n):
count = 0

for i in range(1, n + 1):
if count_divisors(i) == 8:
count += 1

return count

result = f(10**12)
print(result)
“`

This code first defines a helper function `count_divisors` that takes an input `n` and returns the count of divisors of `n`. It iterates from 1 to the square root of `n` and checks if the number is divisible. If it is, it increments the count. Additionally, if the result of dividing `n` by the current iteration is not equal to the current iteration itself, it also increments the count.

The main function `f(n)` iterates from 1 to `n` and checks if the count of divisors for each number is exactly eight. If it is, it increments the count variable. Finally, it returns the counted value.

In the last section, we call `f(10**12)` to find the value of $f(10^{12})$ and print the result.

Please note that calculating $f(10^{12})$ using this code may take a significant amount of time due to the large input value.

More Answers: