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Totient Stairstep Sequences

Problem 337

Published on Saturday, 7th May 2011, 10:00 pm; Solved by 327

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+1 1

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