Sum of Distinct Values of $AFFLUENCE_p(k)$

As simple the problem is its difficulty comes from its poor formulation. "For ," in second sentence looks superfluous. Term "Natural numbers" is ambiguous as sometimes 0 is counted as natural number too. I don't know what should mean "function assumes". I have translated it as "function gives finite set of (resulting) values". It would be very nice if the problem formulation could be corrected somehow. It is fine problem, it resembles problems from our math exams.

Solution:

As cryptic the notation " $p^k|n!$ and $p^{k+1}\not| n!$ " can look it doesn't mean anything else than $k$ in $p^k$ is power of $p$ in factorization of $n!$ .

Example for $p=3$ and $n=28$ :

$28! = 2^{25} \cdot 3^{13} \cdot 5^6 ...$

thus $k = 13$ .

Now, look at how $k$ changes for given $p$ for subsequent values of $n$ .

Example for $p=3$ :

We can observe that only $n$ which are multiples of $p$ increase value of $k$ and in opposite those multiples always increase it. Including the fact that $k$ cannot decrease with increasing $n$ we can conclude that the sequence of $n$ with same value of $k$ is always $p$ numbers long.

Thus, $AFFLUENCE_p(k)=p$ for $k>0$ .

The only problem is how long is the sequence of $n$ for $k=0$ , because it depends on our definition of "Natural numbers":

$AFFLUENCE_p(0) = p$ if we count 0 as natural number, or

$AFFLUENCE_p(0) = p-1$ if we don't.

Thus, the result is $p$ or $p + (p-1)$ .