[seqfan] Re: Two journeys through the positive integers

[M. F. Hasler]

On Wed, Dec 4, 2019 at 3:31 AM Éric Angelini <bk263401 at skynet.be> wrote:

> 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, ,...
>
> 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?


Eric, I told you that it is not.
Specifically, the a(5) = 9 must be a 2 in the lex.earliest version, see
https://oeis.org/draft/A330154.
You told me that your journey is computed by choosing *at each step* (along
the journey, not for indices n=1,2,3...)
the *largest* possible prime or, if not possible, the smallest possible
composite.
I think we can't even expect that a rule saying "choose the largest
possible" would yield the lex.earliest sequence....

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".
>

The number of primes is equal to that of composite numbers.
They may/will "lag behind" but that's completely irrelevant.
(Actually, the lower density of the primes is even somewhat compensated by
the fact that they are used less frequently along your journey.)

-- 
Maximilian

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