Why is $a_{214}$ nonintegral where $a_0 = 1, a_{n+1}=\frac{1 + a_0^5 + \cdots + a_n^5}{n + 1}?$ Define the sequence $a_n$ by the following. 

$$a_0 = 1,$$
$$a_{n+1}=\frac{1 + a_0^5 + \cdots + a_n^5}{n + 1}$$

$a_{214}$ is nonintegral.
（https://oeis.org/A108394）
Please tell me the proof.
(By computing the sequence modulo 251 it is easy to show that $a_{251}$ is not integral.)

EDIT: This isn't fully correct and contains some imprecise calculations. The core is using modulo 215. To misquote the OP: "By computing the sequence modulo 215 it is easy to show that $a_{215}$ is not integral."
ORIGINAL:
Doing the calculations modulo 215 gives the following:
$$a_0=1$$
$$a_1=\frac{1+1^5}{1}=2$$
$$a_2=\frac{1+1^5+2^5}{2}=17$$
$$a_3=\frac{1+1^5+2^5+17^5}{3}=473297\equiv82\pmod{215}$$
$$a_4\equiv\frac{1+1^5+2^5+17^5+82^5}{4}\equiv42\pmod{215}$$
$$a_5\equiv\frac{1+1^5+2^5+17^5+82^5+42^5}{5}\equiv171\pmod{215}$$
$$a_6\equiv\frac{1+1^5+2^5+17^5+82^5+42^5+171^5}{6}\equiv171\pmod{215}$$
All consecutive terms are 171 module 215 which leads to your conclusion.
