# Another silly digit-based sequence

David Wilson davidwwilson at comcast.net
Sat Apr 23 22:54:45 CEST 2005

```It appears to me, that for any string of digits s, there is a minimal
number f(s) such that for any integer k > 0, s appears as a substring
of some element of the set {xk | 1 <= x <= f(s)}.

For example, for any k > 0, the string 0 appears in one of k, 2k,
3k, 4k, ..., 10k (specifically, it appears at the end of 10k), so f(0) <=
10.
Letting k = 1 shows f(0) >= 10, so f(0) = 10.

Similarly, for any k > 0, it looks as if 1 appears in one of k, 2k, 3k, 4k
or 5k.
k = 2 shows f(s) >= 5.

It looks as if f(s) exists for every string of digits s.  If we take s = 0,
1, 2,...
it looks as if f(s) starts off

10, 5, 12, 17, 32, 25, 24, 35, 32, 72, 81, 101, 111, 111, ...

I sure I could find and prove the correct values if I had the appropriate
tools for dealing with finite automata.

- David W. Wilson

"Truth is just truth -- You can't have opinions about the truth."
- Peter Schickele, from P.D.Q. Bach's oratorio "The Seasonings"

```