Some basic permutations

Let \(f(n)\) equal the average over all permutations of the digits \(1\) to \(n\) (inclusive) interpreted in base \(n+1\). For example, for \(f(3)\) we have

\[(123)_4 = (27)_{10}\] \[(132)_4 = (30)_{10}\] \[(213)_4 = (39)_{10}\] \[(231)_4 = (45)_{10}\] \[(312)_4 = (54)_{10}\] \[(321)_4 = (57)_{10}\]

for an average value of \(42\) (in base 10).

Find \(f(10^6)\) mod \(1000037\).

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