Daniel's path sums

Probability Level 3

For integers $n > 2$ , let $G_n$ be a complete graph on $n$ vertices such that each vertex is labeled by a distinct number $1,2,3,\cdots,n$ , and each edge is labeled by the sum of its endpoint labels. Let $f(G_n)$ be the minimum sum of edge labels in any path that touches every vertex in $G_n$ exactly once.

How many values of $n$ satisfy $f(G_n)\equiv 2013\pmod{n}?$