[seqfan] Jumping back and forth (on even and odd terms)

[Eric Angelini]

Hello SeqFans,
The hereunder seq U is a nice permutation of the Naturals, I guess:
U = 1,5,3,9,7,23,4,2,13,17,19,27,8,11,25,21,15,31,29,35,39,41,12,37,33,16,47,10,6,18,14,49,26,45,51,53,...

The idea is to start on a(1)=1 and from there:
-- if a(n) is odd, then jump to the right over a(n) terms;
-- if a(n) is even, then jump to the left over a(n) terms;
-- the term you land on will be the starting point for the next jump.

I think U is the lexicographically first such seq if, in extending U, one always tries to fill the leftmost "hole" with an integer not yet present in U and not leading to a contradiction.

I'm currently computing by hand the seq P where you jump to the right if a(n) is not a prime and to the left if a(n) is prime. P starts:
P = 1,6,4,10,8,9,12,5,3,18,15,...

Best,
É.