# What digits appear in 2,3,6,1,8,6,8,4,8,4,8…?

The sequence begins: $2,3,6,1,8,6,8,4,8,4,8....$

(See OEIS A093095.)

$2*3=6; 3*6=1,8; 6*1=6; 1*8=8; 8*6=4,8;$ and so on.

Will there ever be a $5$? Will the sequence ever repeat?

I tried doing this by hand, and so far the only numbers I have are $1,2,3,4,6,8$: none of which can be multiplied together to get a $5$.

Which digits never occur? How does one prove this in general?

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Since you are always taking the product of single digit numbers, you might want to pull out a multiplication table: Almost the only time $m*n$ for $m,n$ single digits produces a $0, 5, 7$ or $9$ is when one (or both) of $m$ and $n$ includes a $0, 5, 7$ or $9$.
The one exception is $3*3 = 9$, but no single digit numbers multiply to give $33$.
Thus, $0, 5, 7$ and $9$ never appear in this sequence.