# Base-10 Skolem-Langford numbers

Eric Angelini Eric.Angelini at kntv.be
Mon Aug 27 16:23:55 CEST 2007

```
... a nice (?) seq would be the seq of:

"First Skolem-Langford numbers having n as substring"
The fisrt SL with the substring 2007, for instance, seems
to be 4712142007

Seq would start:
131003, 2002, 131003, 420024, 15120025, 61310036, etc.
Seq is finite.

Another nice (?) seq would be:

"Integers n which never appear as a substring in a SL number"

Seq starts (I think) like this:
11, 22, 33, 44, 55, 66, 77, 88, 99, 101, etc.
Seq is infinite.

Best,
É.

On Aug 27, 2007, at 9:28 AM, Eric Angelini wrote:

> ... are now here, thanks to David Wilson:
> http://www.research.att.com/~njas/sequences/A108116

Thanks Eric, And thank you David. While my own compute of this
Wilson's result.

[By the way, Neil, the link to David's 20120 b-file only lists 33.]

Because the method I used to compute the length-18 and length-16
numbers involved separate sub-routines for which digits were missing,
I noticed the following:

The conjecture is trivially true in the sense that I can verify it
for all 20120 terms. However, I wonder if someone has a simple proof
of it. It would immediately follow from that proof that no length-20
terms exist (because half the sum of its digits is 45) and it would
provide insight into the facts that the missing digit in length-18
terms is odd and that the two missing digits in length-16 terms are
of opposite even-odd parity.

Hans

On Aug 27, 2007, at 10:23 AM, Eric Angelini wrote:

> "First Skolem-Langford numbers having n as substring"
> Seq would start:
> 131003, 2002, 131003, 420024, 15120025, 61310036, etc.
> Seq is finite.

LOL. I actually programmed this. But as you pointed out subsequently:

> "Integers n which never appear as a substring in a SL number"
> Seq starts (I think) like this:
> 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, etc.

... so 11 never appears. Only 10 terms! Hardly worth the program. :)
Of course, one could put a zero where the substring does not appear,
but this would make the sequence infinite (with only zeros after term
#978416154798652002).

Hans

```