MathDB

For any nonnegative integer

r

, let

S_r

be a function whose domain is the natural numbers that satisfies

S_r(p^{\alpha}) = \begin{cases} 0\,\, if \,\, if \,\, p \le r \\ p^{{\alpha}-1}(p -r) \,\, if \,\,p > r \end{cases}

for all primes

p

and positive integers

{\alpha}

, and that

S_r(ab) = S_r(a)Sr_(b)

whenever

a

and

b

are relatively prime. Now, suppose there are

n

squirrels at a party. Each squirrel is labeled with a unique number from the set

\{1, 2,..., n\}

. Two squirrels are friends with each other if and only if the difference between their labels is relatively prime to

n

. For example, if

n = 10

, then the squirrels with labels

3

and

10

are friends with each other because

10 - 3 = 7

, and

7

is relatively prime to

10

. Fix a positive integer

m

. Define a clique of size

m

to be any set of m squirrels at the party with the property that any two squirrels in the clique are friends with each other. Determine, with proof, a formula (using

S_r

) for the number of cliques of size

m

at the squirrel party.

Problem Statement