[seqfan] Two journeys through the positive integers

[Éric Angelini]

Hello SeqFans,
Start on a(1) = 1 and obey this simple rule: "If a(n) is not a prime, skip a(n) places to the right, else to the left".

1, 4, 6, 8, 9, 3, 12, 14, 5, 21, 25, 7, 20, 10, 51, 16, 26, 22, 11, 15, 34, 2, 28, 17, 63, 42, 48, 33, 36, 32, 13, 18, 23, 60, 19, 24, 52, 91, 45, 29, 69, 55, 27, 39, 66, 58, 116, 78, 85, 37, 30, 105, 94, 90, 35, 75, 81, 77, 112, 43, 31, 41, 65, 106, 38, 50, 72, 44, 141, 47, 133, 180, 100, 150, 49, 121, 168, 128, 110, 123, 53, 146, 40, 46, 159, 162, 98, 64, 54, 61, 140,...

We see here that a(1) = 1 drives you to 4; 4 drives you to 3; 3 brings you back to 6; 6 drives you to 5; 5 brings you back to 8; 8 drives you to 7; 7 brings you back to 9; 9 drives you to 10; 10 drives you to 17; etc.

This should be the lexicographically earliest infinite sequence S of distinct positive integers that visits only once every term of S. But is it sure? Is S infinite? As Hugo P. wrote on the sequence's draft, "Shall every position be visited? Seems to be difficult given the abundance of composites against the primes".

A beautiful variant (in Jean-Marc Falcoz and Eric's eyes :-) is shown there, on my private blog:
https://bit.ly/2OKsKxz
Best,
É.