## Let $S(n) = \sum a + b + c$ over all triples $(a, b, c)$ such that:

$a$, $b$ and $c$ are prime numbers.

$a \lt b \lt c \lt n$.

$a+1$, $b+1$, and $c+1$ form a geometric sequence.

For example, $S(100) = 1035$ with the following triples:

$(2, 5, 11)$, $(2, 11, 47)$, $(5, 11, 23)$, $(5, 17, 53)$, $(7, 11, 17)$, $(7, 23, 71)$, $(11, 23, 47)$, $(17, 23, 31)$, $(17, 41, 97)$, $(31, 47, 71)$, $(71, 83, 97)$

Find $S(10^8)$.

### To find the expected distance from the depot for a given value of n, we can simulate the scenario using Python code. Here’s the approach we’ll take:

1. We’ll iterate over each second until only one drone is left at the depot.

2. At each second, we’ll randomly select a drone and update its speed or position based on its current state.

3. We’ll keep track of the total distance covered by all the drones so far.

4. Once only one drone is left at the depot, we’ll calculate the average distance covered by each drone by dividing the total distance by n.

Here’s the Python code to solve this problem:

“`python

import random

def expected_distance(n):

total_distance = 0

depot = [0] * n

while len(depot) > 1:

drone = random.choice(depot)

if drone == 0:

depot.append(1) # Start moving at 1 cm/s

else:

depot.append(drone + 1) # Increase speed by 1 cm/s

depot.remove(drone)

total_distance += drone # Add distance covered by the selected drone

return round(total_distance / n)

result = expected_distance(10**8)

print(result)

“`

When we run this code, it will simulate the scenario with 10^8 drones and calculate the rounded average distance covered by each drone. This may take some time to run due to the large number of iterations involved.

Note: The `random.choice` function selects a random element from a list. We use it here to randomly select a drone from the depot at each second.

Please note that the output may vary slightly between runs due to the random nature of the problem.

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