[seqfan] Fractal with absolute differences?

[Éric Angelini]

Hello SeqFans,
I would like to introduce a new (?) concept of fractal sequence.
The idea is to select some terms of a seq (the primes, for instance) and to coin a rule that would involve only those terms. The mix "selected terms" + "rule" would reproduce the seq itself.
Consider for instance the primes in S:
S = 1,2,4,6,8,9,5,11,3,12,7,18,10,13,25,14,15,16,...
We highlight them:
S = 1,(2),4,6,8,9,(5),(11),(3),12,(7),18,10,(13),25,14,15,16,...
Now compute for each prime p of S the absolute differences |p-L| and |p-R|, with L being the term on the immediate left of p, and R the term on the immediate right of p.
The prime 2 would then produce the absolute differences 1 and 2.
If we gather all such differences, we will reproduce S. This is the idea. We would like of course to find the lexico-first S of distinct terms > 0 having this property. Will S be finite? Will it block itself at some point? The author doesn't know and would appreciate some help.
More on my blog: https://bit.ly/38HwVSr
Best,
É.