Ruff Numbers

Let $S_k$ be the set containing $2$ and $5$ and the first $k$ primes that end in $7$. For example, $S_3 = \{2,5,7,17,37\}$.

Define a $k$-Ruff number to be one that is not divisible by any element in $S_k$.

If $N_k$ is the product of the numbers in $S_k$ then define $F(k)$ to be the sum of all $k$-Ruff numbers less than $N_k$ that have last digit $7$. You are given $F(3) = 76101452$.

Find $F(97)$, give your answer modulo $1\,000\,000\,007$.

To solve this problem, we can use Python to generate the set $S_k$, find the $k$-Ruff numbers, calculate their sum, and finally take the result modulo $1\,000\,000\,007$.

Let’s break down the steps to solve this problem:

Step 1: Generate the set $S_k$
To generate the set $S_k$, we need to find the first $k$ primes that end in $7$. We can use the `sympy` library in Python to generate prime numbers. Install the `sympy` library if you haven’t already, by running the command `pip install sympy`.

“`python
from sympy import isprime

def generate_S(k):
primes = [7] # Start with 7 since it is already known
n = 11 # First prime number ending in 7

while len(primes) < k: if str(n)[-1] == '7' and isprime(n): primes.append(n) n += 10 # Only check numbers ending in 7 return set(primes) ``` Step 2: Find the $k$-Ruff numbers To find the $k$-Ruff numbers, we need to consider all possible numbers less than the product of $S_k$ such that they are not divisible by any element in $S_k`$ and have the last digit 7. ```python def find_k_ruff_numbers(S): max_number = max(S) k_ruff_numbers = [] for num in range(7, max_number, 10): if all(num % prime != 0 for prime in S): k_ruff_numbers.append(num) return k_ruff_numbers ``` Step 3: Calculate the sum of $k$-Ruff numbers Once we have the list of $k$-Ruff numbers, we can calculate their sum. ```python def calculate_sum_of_k_ruff_numbers(k_ruff_numbers): return sum(k_ruff_numbers) ``` Step 4: Calculate $F(k)$ modulo $1\,000\,000\,007$ Finally, we can calculate $F(k)$ modulo $1\,000\,000\,007$. ```python def calculate_F_k(k): S = generate_S(k) k_ruff_numbers = find_k_ruff_numbers(S) sum_of_k_ruff_numbers = calculate_sum_of_k_ruff_numbers(k_ruff_numbers) return sum_of_k_ruff_numbers % 1000000007 ``` Now, let's calculate $F(97)$ using the defined functions: ```python F_97 = calculate_F_k(97) print(F_97) ``` The output will be the value of $F(97)$ modulo $1\,000\,000\,007$.

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