# [seqfan] Arithmetic progressions such that adjacent terms have a common non-zero digit

Sun May 26 19:49:33 CEST 2019

```Daniel Griller asked the following question:

Does there exist a positive integer n such that every term in the
sequence n, 2n, 3n, 4n, 5n, ... has a non-zero digit in common with the
next term?

If we allowed 0 as a common digit then every multiple of 10 would be a
solution.

I was able to prove the following:

- 99 is the smallest positive integer with this property.
- 10^k - 1 has this property for every k > 1.
- A positive integer n has this property if and only if 10*n has this
property.

Note that the adjacent terms k*n and (k+1)*n usually have the same leading
digit, and it suffices to look at the boundaries where the leading digit
changes.

The sequence of numbers having this property might begin as follows, but I
have not proved this.

99, 990, 999, 1998, 2997, 3996, 3999, 4992, 4995, 5994, 6875, 6993, 6996,
7992, 8125, 8704, 8991, 9856, 9900, 9984, 9990, 9999

Is there an efficient algorithm to generate the terms of this sequence?

```