Göbel's sequence

This sequence was developed by the German mathematician Fritz Göbel in the 1970s.^[2] In 1975, the Dutch mathematician Hendrik Lenstra showed that the 44th term is not an integer.^[2]

Göbel's sequence can be generalized to kth powers by

${\displaystyle x_{n}={\frac {1+x_{0}^{k}+x_{1}^{k}+\cdots +x_{n-1}^{k}}{n}},}$

with starting value

${\displaystyle x_{0}=1.}$

The least indices at which the k-Göbel sequences assume a non-integral value are

43, 89, 97, 214, 19, 239, 37, 79, 83, 239, ... (sequence A108394 in the OEIS)

Regardless of the value chosen for k, the initial 19 terms are always integers.^[3][2]

• Somos sequence