Observe that is well defined for since has spanning trees.
The function was introduced by Sedlacek [S] who has shown that for large enough and
Using the fact that almost all positive integers are expressible as for integers it can be shown [A] that for large enough
Moreover, the only fixed points of are 3, 4, 5, 6, 7, 10, 13 and 22.
The conjecture is motivated by the following graph (ploted for a very small sample of vertices)
The conjecture [C] is justifiable for highly composite numbers since in this case one can construct the graph obtained after taking cycles for every odd prime factor of .
[S] J. Sedlacek, On the minimal graph with a given number of spanning trees, Canad. Math. Bull. 13 (1970) 515-517.
[A] J. Azarija, R. Skrekovski, Euler's idoneal numbers and an inequality concerning minimal graphs with a prescribed number of spanning trees, IMFM preprints 49 (2011) Link to paper
* indicates original appearance(s) of problem.