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

M. F. Hasler seqfan at hasler.fr
Tue May 28 20:01:14 CEST 2019

```On Tue, May 28, 2019, 12:03 Benoît Jubin wrote:

> >
> > I think we agreed that terms with trailing 0 should be excluded, since
> > these are trivial solutions.
>
> Why should trivial solutions be excluded by principle (not only for this
> sequence, but in general)?  I think that if one wants a sequence of
> "nontrivial solutions", this should be a second auxiliary sequence.
>

I agree, there should always be both, the "full" sequence as well as that
of "primitive terms" which easily allow to (re)construct the full sequence.
Traditionally the second was preferred because the density of information
is higher, i.e., it allows to show more "interesting" terms within the 3
lines of data.

> > >    - A positive integer n has this property if and only if 10*n
> has this  property.
> >
> > I disagree with the last one, since for ANY integer n > 0, 10n has
> the required property.
>
> You may have overlooked the requirement that multiples share a *nonzero*
> digit.
>

Oh yes indeed, you do well to remind this.
Is it easy to see whether or not this changes anything for the subsequence
of primitive terms (not multiples of 10)?

By the way, which digit it is (or which they are) could also be
> interesting.  It looks like it will most often be 9.

I think asymptotically all digits are equally often among the common digits.
Because what happens at the limits m*10^p I considered earlier is
"negligible", and generically consecutive terms will have almost all of
their digits in common.
(And almost all sufficiently large terms have a roughly equal nonzero
number of each digit.)

looking at the situation in smaller bases could be interesting
>

I totally agree, in this and (almost) all other "digit" related sequences.
It is extremely rare that 10 has anything particular that would make it
special and mathematically justify a "privileged" treatment (as opposed to
bases which are primes or powers of primes, especially of 2, for example).

- Maximilian

```