[seqfan] A fractal seq with (primes by concatenation) pairs

[Eric Angelini]

Hello SeqFans,

Pr = 0,2,0,0,3,2,0,5,0,0,7,0,1,3,3,2,3,2,1,1,0,1,7,5,1,9,0,4,1,0,4
,7,7,2,9,0,8,9,1,4,3,3,5,3,3,3,1,2,5,9,3,3,7,2,7,9,1,10,9,1,6,1,0,
1,37,1,6,7,7,7,1,5,1,39,1,8,3,9,9,7,0,10,1,4,14,9,1,10,3,0,10,7,4,
12,7,7,15,7,7,7,3,2,1,31,9,1,13,0,1,51,8,1,63,9,1,67,1,17,3,4,17,9,
3,19,3,3,18,1,5,1,91,3,19,9,3,22,3,3,2,11,1,19,7,2,2,27,5,2,29,9,2,
33,3,2,39,3,2,41,7,2,51,2,25,7,7,2,63,9,26,9,1,2,71,10,2,77,9,2,83,
1,2,81,6,...

Erase all pairs of terms forming a prime by concatenation; 
the remaining terms will reproduce the original sequence:

Pr = (0,2),0,(0,3),2,(0,5),0,(0,7),0,(1,3),3,(2,3),2,(1,1),0,(1,7),5,(1,9),0,...

We consider here that (0,2), for instance, represents the prime "2"
and (0,3) the prime "3", etc. We see that the 5th "prime" (1,3) is 
not (1,1); this is because (1,1) would have produced a "prime-overlap"
with the following "3" : (1,1),3, --and we don't want such overlaps. 
So we took the next available "split-prime" to extend the sequence,
which is "13".
After "13" we could still not use "11" -- as we would have another
overlap, but just before "11": 3,(1,1),2, -- indeed, 31 is prime.
So we tried 17 and 19 (same problem) before choosing "23". 

How was Pr build? Pr is a succession of triplets s, t, u:

s t u
0,2,0,
0,3,2,
0,5,0,
0,7,0,
1,3,3,
2,3,2,
1,1,0,
1,7,5,
1,9,0,
4,1,0,
4,7,7,
2,9,0,
8,9,1,
4,3,3,
5,3,3,
3,1,2,
5,9,3,
3,7,2,
7,9,1,
10,9,1,
...

-- s concatenated with t always produces a prime (s,t);
-- this (s,t) prime is always the smallest not yet present in Pr;
-- the term just before s, concatenated to s, never produces a prime;
-- t and u, concatenated, never produce a prime;
-- there is no choice for u, because u is given by the seq itself
   (as we want Pr to be "fractal)

Best,
É.