Non-Integral Value of 3-Göbel Sequence
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Theorem
Consider the $3$-Göbel sequence defined recursively as:
- $x_n = \begin {cases} 1 & : n = 0 \\ \ds \paren {1 + \sum_{k \mathop = 0}^{n - 1} {x_k}^3} / n & : n > 0 \end {cases}$
The smallest $n$ such that $x_n$ is not an integer is $88$.
Proof
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Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $43$
- Weisstein, Eric W. "Göbel's Sequence." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GoebelsSequence.html