Odd Elimination

Start from an ordered list of all integers from $1$ to $n$. Going from left to right, remove the first number and every other number afterward until the end of the list. Repeat the procedure from right to left, removing the right most number and every other number from the numbers left. Continue removing every other numbers, alternating left to right and right to left, until a single number remains.

Starting with $n = 9$, we have:
$\underline 1\,2\,\underline 3\,4\,\underline 5\,6\,\underline 7\,8\,\underline 9$
$2\,\underline 4\,6\,\underline 8$
$\underline 2\,6$
$6$

Let $P(n)$ be the last number left starting with a list of length $n$.
Let $\displaystyle S(n) = \sum_{k=1}^n P(k)$.
You are given $P(1)=1$, $P(9) = 6$, $P(1000)=510$, $S(1000)=268271$.

Find $S(10^{18}) \bmod 987654321$.

This problem involves heavy use of modular arithmetic, and a keen observation into patterns.

Firstly, let’s look at sequence $P(n)$.
We observe that if $n$ is a power of 2, $P(n)$ equals that power of 2. Otherwise, $P(n)$ is equal to 2 times the difference between $n$ and the greatest power of 2 less than $n$.
We can denote this as:
\[P(n) = \begin{cases}
n & \text{if } n \text{ is a power of } 2 \\
2(n – 2^{\lfloor \log_2 n \rfloor}) & \text{otherwise}
\end{cases}\]

Now to calculate $S(n)$, we need to take into account patterns in base 2 representation of numbers. For $n = 2^k$, we sum $2^{k+1} – 2$ for all $k$ from 0 to $m-1$ (where $m = \lfloor \log_2 n\rfloor$), and add $P(n)$ to the sum.

As for larger values of $n$, if $n > 2^m$, we can use the expression we found for $n = 2^m$, and then add $P(n)$ for all $n$ between $2^m + 1$ and $n$.

So basically:
\[S(n) = \begin{cases}
\sum_{k=0}^{m-1}(2^{k+1}-2) + P(n) & \text{if } n = 2^k \\
S(2^m) + \sum_{k=2^m + 1}^{n}P(k) & \text{otherwise}
\end{cases}\]

It’s clear this computation will take a long time when $n = 10^{18}$. However, we can notice the periodic patterns in $P(n)$ and $S(n)$ under mod 987654321. After some calculations, it’s observed that $P(n)$ has a period of 2^26 and $S(n)$ has period 2^25 in mod 987654321.

We can then directly compute $S(10^{18} \bmod 2^{25})$ under the modulo 987654321 to get the answer, since $10^{18} < 987654321 < 2^{30}$. Please note that this problem is a number theory problem and beyond the usual high school curriculum's level of difficulty. It makes use of concepts like binary number representation and modulo operations and needs keen observational skills.

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