Daniel's path sums

For integers n>2n > 2, let GnG_n be a complete graph on nn vertices such that each vertex is labeled by a distinct number 1,2,3,,n1,2,3,\cdots,n, and each edge is labeled by the sum of its endpoint labels. Let f(Gn)f(G_n) be the minimum sum of edge labels in any path that touches every vertex in GnG_n exactly once.

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

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