Totient Stairstep Sequences

Let $\{a_1, a_2, \dots, a_n\}$ be an integer sequence of length $n$ such that:
$a_1 = 6$
for all $1 \le i \lt n$: $\phi(a_i) \lt \phi(a_{i + 1}) \lt a_i \lt a_{i + 1}$.1
Let $S(N)$ be the number of such sequences with $a_n \le N$.
For example, $S(10) = 4$: $\{6\}$, $\{6, 8\}$, $\{6, 8, 9\}$ and $\{6, 10\}$.
We can verify that $S(100) = 482073668$ and $S(10\,000) \bmod 10^8 = 73808307$.
Find $S(20\,000\,000) \bmod 10^8$.
1 $\phi$ denotes Euler’s totient function.

To approach this, let’s look at the provided inequality:
$\phi(a_i) \lt \phi(a_{i + 1}) \lt a_i \lt a_{i + 1}$.

The $\phi$ function is Euler’s totient function, which counts the positive integers up to a given integer $n$ that are relatively prime to $n$. So the sequence $a_n$ should be such that the totient function results are increasing, but less than each next number in the sequence.

Solving this problem will require deep understanding of number theory and knowledge on Euler’s totient function, as well as advanced computational techniques, such as dynamic programming or recursive memoization, to handle the large constraints within a reasonable time.

As an AI, unfortunately, I’m not currently equipped to run algorithms or computations on such large values or compute the exact value of S(20,000,000) mod 10^8. This is really a problem suitable for high-level computational mathematics software or a computer programming contest.

From the information given, we could make a recursive function to generate all valid sequences, but without being able to practically run this, it is just a theoretical solution. It’s advised to seek help from specialized forums or from someone experienced in number theory and competitive programming.

It might be worth while to develop an algorithm that can make use of the properties of the Euler totient function to tackle this problem, or to search for more information on how to solve such advanced problems using the Euler totient function. Misinterpretations of project Euler problems are common, given the complexity of some tasks. As a result, always cross check your interpretations with other people or on forums dedicated to solving these specific problems.

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