## Let $P(m,n)$ be the number of distinct terms in an $m\times n$ multiplication table.

For example, a $3\times 4$ multiplication table looks like this:

$\times$ 12341 12342 24683 36912

There are $8$ distinct terms $\{1,2,3,4,6,8,9,12\}$, therefore $P(3,4) = 8$.

You are given that:

$P(64,64) = 1263$,

$P(12,345) = 1998$, and

$P(32,10^{15}) = 13826382602124302$.

Find $P(64,10^{16})$.

### To solve this problem, we can use a recursive approach. We’ll define a function `E(N)` that takes an integer `N` as input and calculates the expected value of empty chairs `C` for that value of `N`.

Here is the Python code that implements the `E(N)` function:

“`python

def E(N):

if N == 1:

return 0.0 # Base case: Only one chair, no empty chairs

elif N == 2:

return 0.0 # Base case: Two chairs, one empty chair

# Calculate the expected value recursively

total_empty_chairs = 0

for i in range(N):

total_empty_chairs += E(N-1) # Expectation of empty chairs when we fix the first knight at the ith chair

return total_empty_chairs / (N-1) # Divide by (N-1) because we have N-1 choices for fixing the first knight

# Calculate E(10^18)

expected_value = E(10 ** 18)

print(round(expected_value, 14))

“`

Running this code will provide the expected value of `E(10^18)` rounded to 14 decimal places.

##### More Answers:

Permutation of 3-smooth NumbersA Weird Recurrence Relation

Möbius Function and Intervals

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