Pound notation in Summation
In the third paragraph of the first section (page 1), they define $\#(w)$, $\#(w; c)$, and $\#(c)$. Quoting such paragraph:


  Word-context pairs are denoted as $(w; c)$, and $\#(w; c)$ counts all observations of $(w; c)$ from the corpus. We use $\#( w ) =  \sum_c \#(w; c)$ and $\#(c) = \sum_w \#(w; c)$ to refer to the count of occurrences of a word (context) in all word-context pairs. Either $\sum_c \#(c)$ or  $\sum_w \#(w)$ may represent the count of all word-context pairs.


So, considering the equation you cited:


$\#(w_i, c_j)$ counts the occurrences of a specific word $w_i$ over the context $c_j$;
and $\frac{\#(c_)}{\sum_w \#(w)}$ accounts for the the probability of having context $c_j$ in the dataset.