Problem 337: Totient Stairstep Sequences

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Let {a1, a2,..., an} be an integer sequence of length n such that:

a1 = 6

for all 1 ≤ i < n : φ(ai) < φ(ai+1) < ai < ai+11

Let S(N) be the number of such sequences with an ≤ N.

For example, S(10) = 4: {6}, {6, 8}, {6, 8, 9} and {6, 10}.

We can verify that S(100) = 482073668 and S(10 000) mod 108 = 73808307.

Find S(20 000 000) mod 108.

1 φ denotes Euler's totient function.