## There are $N$ seats in a row. $N$ people come one after another to fill the seats according to the following rules:

No person sits beside another.

The first person chooses any seat.

Each subsequent person chooses the seat furthest from anyone else already seated, as long as it does not violate rule 1. If there is more than one choice satisfying this condition, then the person chooses the leftmost choice.

Note that due to rule 1, some seats will surely be left unoccupied, and the maximum number of people that can be seated is less than $N$ (for $N \gt 1$).

Here are the possible seating arrangements for $N = 15$:

We see that if the first person chooses correctly, the $15$ seats can seat up to $7$ people.

We can also see that the first person has $9$ choices to maximize the number of people that may be seated.

Let $f(N)$ be the number of choices the first person has to maximize the number of occupants for $N$ seats in a row. Thus, $f(1) = 1$, $f(15) = 9$, $f(20) = 6$, and $f(500) = 16$.

Also, $\sum f(N) = 83$ for $1 \le N \le 20$ and $\sum f(N) = 13343$ for $1 \le N \le 500$.

Find $\sum f(N)$ for $1 \le N \le 10^{12}$. Give the last $8$ digits of your answer.

### 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.

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Empty ChairsSuper Ramvok

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