A Sum Of Products Of Sequences

\[ \begin{cases} x_{n}=(x_{n-1}+1)^{2}+2 \left \{ \frac{n}{2} \right \}\quad (n\ge 2) \\ x_{1}=4 .\end{cases} \] Define \(x_{n}\) as an infinite sequence of numbers, as shown above.

Define \(y_{n}\) as an infinite sequence of non-negative integers such that \(n|y_{n}\) for all \(n\ge 1\).

What is the largest integer that cannot be expressed in the form \[\sum_{n=1}^{j}x_{n}y_{n}\] for some positive integer \(j\)?

Notations:

  • \(n|y_{n}\) means that \(n\) divides \(y_{n}\). Note that \(n|0\) for \(n \neq 0 \).
  • \(\{\cdot \}\) denotes the fractional part function. For example, \(\{ 12.3345 \}=0.3345\).
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