# [seqfan] |a-b| divides concatenation [ab]

Eric Angelini Eric.Angelini at kntv.be
Mon Sep 20 18:15:15 CEST 2010

```Hello SeqFans,

S = 1,144,146,153,156,160,165,176,184,197,274,288,294,315,324,336,352,374, ...

We want:
1) S to be strictly increasing
2) all first diff to be different one from another and not yet present in S
3) a(n+1) to be the smallest integer such that |a(n)-a(n+1)| divides the
concatenation [a(n),a(n+1)]

Here is how we get S, starting with 1:

S = 1,144,146,153,156,160,165,176,184,197,274,288,294,315,324,336,352,374, ...
1st dif:  143  2   7   3   4   5  11   8  13  77  14   6   21  9  12  16   22

143 is the smallest integer not yet present and dividing 1144   (=8)
2 is the smallest integer not yet present and dividing 144146 (=72073)
7 is the smallest integer not yet present and dividing 146153 (=20879)
3 is the smallest integer not yet present and dividing 153156 (=51052)
4 is the smallest integer not yet present and dividing 156160 (=39040)
5 is the smallest integer not yet present and dividing 160165 (=32033)
11 is the smallest integer not yet present and dividing 165176 (=15016)
...

---

If we drop the "strictly increasing" constraint, we'll get T (which
is an incredible nightmare to calculate by hand):

T = 1, 144, 43, 134, 108, 9, 6, 4, 158, ...
1st dif:  143 101  91   26  99  3  2  154 ...

More terms for S & T (if of interest)?

Best,
É.

```