[seqfan] Re: Two journeys through the positive integers

[M. F. Hasler]

>
> On Wed, Dec 4, 2019 at 3:31 AM Éric Angelini 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.
>>
>
In my last message, I cited the lexicographic earliest sequence (now
oeis.org/A330154 = (1, 4, 6, 8, 2, 9, 3, 10, 12, 5...))
for which each term has a successor and a predecessor (except a(1)=1)
corresponding to these rules, and such that no "loops" would occur.
The unpleasant property of this sequence is that the trajectories of the
earliest elements are not connected early.

However, one can construct sequences which coincide with this minimal one
up to a given rank, and then make efforts to connect the so far unconnected
trajectories, following the method used by Eric and Jean-Marc in A329423:
"along the trip" (trajectory of 1), choose the largest possible prime, or
else the smallest possible composite.
That way we can show that there is a "totally connected" sequence which
coincides with the minimal one up to any desired rank,
and therefore comes lexicographically earlier that the initially suggested
variant (1, 4, 6, 8, 9, 3, 12, 14, ...)

For example,
A_10 = [1, 4, 6, 8, 2, 9, 3, 10, 12, 5, 16, 14, 15, 20, 7, 35, 24, 11, 27,
28, 45, 25, 34, 40, 55, 13, 18, 17, 32, 51, 72, 22, 44, 21, 57, 105, 60,
75, 133, 49, 19, 65, 85, 38, 31, 30, 26, 29, 69, 54, 36, 63, 48, 37, 23,
84, 41, 77, 90, 143, 33, 80, 99, 42, 87, 46, 98, 93, 111, 68, 144, 115, 53,
119, 153, 47, 67, 106, 112, 56, 52, 39, 156, 141, 170, 121, 43, 123, 50,
184, 176, 58, 187, 71, 174,...]
yields the following journey:
*a(1)=1 -> a(2)=4 -> a(6)=9 -> a(15)=7 -> a(8)=10 *-> a(18)=11
*-> a(7)=3 -> a(4)=8 -> a(12)=14 *-> a(26)=13 -> a(13)=15 -> a(28)=17 ->
a(11)=16 -> a(27)=18 -> a(45)=31 -> a(14)=20 -> a(34)=21 -> a(55)=23 ->
a(32)=22 -> a(54)=37 -> a(17)=24 -> a(41)=19 -> a(22)=25 -> a(47)=26 ->
a(73)=53 -> a(20)=28 -> a(48)=29 -> a(19)=27 -> a(46)=30 -> a(76)=47 ->
a(29)=32 -> a(61)=33 -> a(94)=71 -> a(23)=34 -> a(57)=41 -> a(16)=35 ->
a(51)=36 -> a(87)=43 -> a(44)=38 -> a(82)=39 -> a(121)=97 -> a(24)=40 ->
a(64)=42 -> a(106)=73 -> a(33)=44 -> a(77)=67
*-> a(10)=5 -> a(5)=2 -> a(3)=6 -> a(9)=12* -> a(21)=45 -> ...

(Even here, with minimality imposed only up to a(10), a few relatively
"early" terms are visited only significantly later,
like, e.g., a(39)=133  which is successor of  a(386)=347, some 100 steps
later in the trajectory.
That said, there might be more "compact" extensions of a(1..10), I did not
try to find the best variant in that sense.)

The lexicographic earliest variant, A330154, is the limit of these
sequences which are minimal up to n = N,
and thereafter constructed using the rules of A329423.
Each of these is conjecturally a permutation of the positive integers and
"pathwise connected"
in the sense that the trajectory of 1 under n -> n +
a(n)*(-1)^isprime(a(n)) covers all indices n >= 1
[which yields, for each of these sequences, two other permutations of the
positive integers,
namely this trajectory [sequence of indices n: (1, 2, 6, 15,...)] and the
corresponding sequence of a(n), (1, 4, 9, ...)].

Unfortunately it is not obvious whether the limit N -> oo will preserve the
"pathwise connectedness".

- 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.
>>
>