Expectations Between Jokers

Seeing that we want to find all possible integer $n$ such that $E$ is an integer, we should find a general formula for $E$ in terms of $n$ .

We can see that by the Law of Symmetry, the $n$ jokers will evenly divide the $52$ -card deck into $n+1$ sections. Each of these sections would thus have $\dfrac{52}{n+1}$ cards in them.

To count the number of cards between the first joker and last joker exclusive, we can use complementary counting. Instead, let's try to count the number of cards before the first joker inclusive, and after the last joker inclusive.

Clearly, the number of cards before the first joker exclusive is $\dfrac{52}{n+1}$ , as said before. Adding in this extra joker gives $\dfrac{52}{n+1}+1$ .

This is exactly the same as the expected number of cards after the last joker inclusive. Thus, the total number of cards in our complementary counting is $2\left(\dfrac{52}{n+1}+1\right)=\dfrac{104}{n+1}+2$ .

The total number of cards is $52+n$ . Thus, the number of cards between the first joker and the last exclusive is $E=(52+n)-\left(\dfrac{104}{n+1}+2\right)=50+n-\dfrac{104}{n+1}$ .

Now we must make $50+n-\dfrac{104}{n+1}$ an integer. Clearly, $50+n$ is always an integer; thus it suffices to make $\dfrac{104}{n+1}$ an integer.

Note that the factors of $104$ are $1,2,4,8,13,26,52,104$ . We can see that by letting $n+1$ equal any of these factors, $E$ turns into an integer.

Thus, $n+1=1,2,4,8,13,26,52,104$ , or $n=0,1,3,7,12,25,51,103$ . Heeding the restriction $n\ge 2$ , we see the possible values of $n$ is $n=3,7,12,25,51,103$ .

Plugging each of these in and solving for $E$ , we get $E=27,44,54,71,99,152$ . Thus, our final answer is $27+44+54+71+99+152=\boxed{447}$ and we are done.