## The arithmetic derivative is defined by

$p^\prime = 1$ for any prime $p$

$(ab)^\prime = a^\prime b + ab^\prime$ for all integers $a, b$ (Leibniz rule)

For example, $20^\prime = 24$.

Find $\sum \operatorname{\mathbf{gcd}}(k,k^\prime)$ for $1 \lt k \le 5 \times 10^{15}$.

Note: $\operatorname{\mathbf{gcd}}(x,y)$ denotes the greatest common divisor of $x$ and $y$.

### To solve this problem, we need to find the smallest positive integer, $A_n$, that satisfies the given modular congruence conditions. We then need to calculate the sum of all primes up to a given number that divide at least one element in the sequence $A$.

We can write a Python program to solve this problem using the following steps:

1. Define a function `is_prime` to check if a number is prime. This function will take a number `n` as input and return `True` if `n` is prime, and `False` otherwise. We can use the trial division method to check for primality.

2. Define a function `smallest_integer` to find the smallest positive integer, `A_n`, that satisfies the given modular congruence conditions. This function will take an integer `n` as input and return `A_n`. The function will use a loop to iteratively check if each integer satisfies the given conditions. To check if an integer satisfies the conditions, we can use the `%` operator to check the remainders.

3. Define a function `sum_of_primes` to calculate the sum of all primes up to a given number that divide at least one element in the sequence `A`. This function will take an integer `n` as input and return the sum. The function will use a loop to iterate through each prime number up to `n` and check if it divides any element in the sequence `A`. If a prime divides an element in the sequence, we will add it to the sum.

4. Finally, we can call the `sum_of_primes` function with the input `n = 300000` to get the desired result.

Here is the Python code that implements the above steps:

“`python

import math

def is_prime(n):

if n <= 1:
return False
if n == 2 or n == 3:
return True
if n % 2 == 0:
return False
for i in range(3, int(math.sqrt(n)) + 1, 2):
if n % i == 0:
return False
return True
def smallest_integer(n):
i = 1
while True:
satisfies_conditions = True
for j in range(1, n+1):
prime = is_prime(j)
if i % j != (prime-1):
satisfies_conditions = False
break
if satisfies_conditions:
return i
i += 1
def sum_of_primes(n):
primes = []
for i in range(2, n+1):
if is_prime(i):
primes.append(i)
sum_of_divisors = 0
for prime in primes:
A = smallest_integer(prime)
if A is not None:
sum_of_divisors += prime
return sum_of_divisors
# Calculate S(300000)
result = sum_of_primes(300000)
print(result)
```
Running this code will output the value of `S(300000)`, which is the sum of all primes up to 300000 that divide at least one element in the sequence `A`.

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