[seqfan] Interspersion/fractal with prime sums

[Eric Angelini]

Hello SeqFans,
 
I guess the sequence hereunder is an "interspersion" or a "fractal"
(please correct me):
 
"a(n)+a(n+1) is prime and all even terms rebuild the sequence itself":
 
S=1,1,2,1,1,2,1,1,4,1,1,2,1,1,6,1,1,4,1,1,10,1,1,2,1,1,12,1,1,6,1,1,16,1,1,4,1,1,18,1,1,10,1,1,22,1,1,2,1,1,28,1,...
 
All [primes minus 1] will appear, of course -- and will be the only 
even terms of the sequence; the building process is quite visible.
Best,
É.