[seqfan] Re: More about Grant Olson's sequence.A347113

mike at vincico.com mike at vincico.com
Wed Nov 17 21:06:44 CET 2021

Hello folks!

I don't post much, but the Olson sequence A347113 is interesting on account of the 3 constraints that confine prime j or k to divisibility, since equality and coprimality are forbidden, while composites are less constrained.

A little background as this approach may differ and ultimately prove gibberish; skip to the arrow "----->" if you just want to know what I think about the striations for k < n. 

Let j=a(n-1)+1 and k=a(n) in a = A347113. I focused concern on j -> k and hence ratio j/k since j is the progenitor of k; I really didn't pay much attention to n/k except in the gross arrangement of the scatterplot. I've done this with other sequences, for example the "paint sprayer" A279818 (http://www.vincico.com/seq/a279818.html).

We can divide the A347113 into 3 tracts regarding the log-log scatterplot (see http://vincico.com/seq/a347113-9.png) to attempt to explain the quasi-linear striations.

The first group, alpha, concerns all k for which j | k (implying j < k) and k is the product of at least 1 p | j and at least 1 q coprime to j, which we call "semicoprime"; we abbreviate this as (j|k AND k◊j) for brevity. 

Beta group includes all k for which (j◊k AND k|j), implying j > k,with k prime and j = pq, p < q. We see the first 3 primes k = p, the rest have k = q. Beta is "state 1", alpha is "state 3". Therefore all primes in A347113 have (j◊k ∧ k|j) or state 1, and all primes j have (j|k ∧ k◊j) or state 3.

(if the weird character I chose, "lozenge", doesn't transfer, it looks like this "<>" but it was a handy html character to use for an abbreviation so I used it. Its form is not important. I based the abbreviations on Knuth's perpendicular symbol for coprime, and the "|" for divisorship.)

All the remaining terms comprise the central gamma group. The "◊" relation is "neutral", meaning that 1 < gcd(j,k) < j, for instance. The other neutral relation is "||" meaning j | k^e for e > 1, which groups together with j|k as "regular", meaning j | k^e for e >= 0. The "◊" and "||" neutral relations pertain only to composite in the first "slot", and they are not necessarily symmetrical, hence there are 8 value "states" in the sequence. Most terms in A347113 are state 0, (j◊k ∧ k◊j).

I use the term "group" to remember that these striations are not sequential, but rather are produced by similar output from a function and are rather echoes from a main source of input.

Neil described the alpha striations, i.e., alpha_m = 2^m * gamma, since prime j | k implies j < k, and the least k that prime j divides is 2j. The striations for k < n are not as easy to understand, as the "beta" space is stunted by the "smallest missing number", while "alpha" has a great degree of freedom.

-----> We are interested in the striations for k < n.

The terms in beta are prime except 1 (a(33)=25, j = 75); the primes are instigated by squarefree semiprime j = pq, p < q. All the primes in A347113 represent the larger divisor q ("greater-beta") except the very first 3 ("lesser beta"). Are there any more "lesser beta" primes?

Beta striations are organized according to prime j/k (see A349406), but in a way that is non-intuitive. For example, there are two striations that pertain to j/k = 3, spaced a magnitude near 2 apart. a(47) = 29 and a(51) = 19, but both have j/k = 3. For j/k = 5 there appear to be 3 striations, and for j/k = 7, there seem to be at least 2. See this log-log scatterplot of 2^16 terms: ( https://oeis.org/A349406/a349406_1.png ) 

I conject (wildly!) that there are (q − 1) striations for j/k = q prime, centered about the striation β-2 that pertains to q = 2, but many of these have not yet appeared given 2^19 terms of a. I put forward this wild conjecture observing the 2 striations pertaining to q = 3, the lower β-3_2 a bit further from striation β-2 than the higher β-3_1. For q = 5, we see β-5_1, β-5_2, and β-5_3, but not β-5_4 (if it exists). Regarding q = 7, perhaps we see β-7_2 and β-7_5. Perhaps some of the allo-striations (like β-5_4) are repressed by "mechanics of the sequence"; why would that be?

I have nothing else to base the conjecture upon but these observations.

If you want to know more about the work I did, see this text link: http://vincico.com/seq/a347113brief.txt. There is a link to further work that was interrupted in September in that brief.

There is a list of my open questions about the sequence in the further work. I'll post them if you like. Some of the questions are answered, i.e., permutation question.

I am willing to share what data I have with anyone who would like to work with it. In the past month I posted a few sequences that convey some of the data I generated to facilitate your exploration of this "NICE" sequence. (At least I think it's nice.)

(I hope I posted this right. Feel free to throw rotten fruit at me if not. Sorry for long post.)

Best regards,

-----Original Message-----
From: SeqFan <seqfan-bounces at list.seqfan.eu> On Behalf Of M. F. Hasler
Sent: Wednesday, 17 November, 2021 11:05
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Subject: [seqfan] Re: More about Grant Olson's sequence.A347113

On Tue, Nov 16, 2021 at 9:36 PM Neil Sloane <njasloane at gmail.com> wrote:

> we know that A347113 is a permutation we can move on to analyzing its 
> graph.

The most interesting features are the huge spikes that appear.

To me, the scatterplot doesn't suggest "spikes" but rather "rays" of different slopes, that mainly  differ by a factor of 2.

But indeed, when a(n) is on the ray with slope ~ 2^m, then a(n-1) is on the ray with slope 2^(m-1), for n > 1.

The main ray has a slope of almost 1 : around n=10^4  the points on that ray have a(n)/n ~ 0.9987 = 1 - 1 / 768.
Unless I err, all composite numbers not equal to even semiprimes (2p) lie on that branch.
(For example, 472 = 94.4% out of the 500 a(n = 10^4-499 .. 10^4) are on that ray.)

All values on the rays with larger slope are even semiprimes, i.e., of the form 2p :

The secondary ray has a slope very close to 2 :
 a(n) / n ~ 2 - 1 / 207 for  a(n)  on that  ray, around n=10^4.
Only about 10% of the a(n) are on that ray, at least near n=10^4 Specifically, 22 among the 200  a(n = 10^4-199 .. 10^4).

Then, there are minor rays with larger slope :
One with slope ~ 4 : a(n) / n = 4 - 1/445 near n=10^4 ; 16  values among
a(9e3 .. 1e4).
One with slope ~ 8 : a(n) / n = 8 - 1/13 near n=10^4 ; only 2 values for
9e3 < n < 1e4.
one with slope ~ 16 : only 7 values a(n), n < 10^4.
The values  a(93) = 2878  and  a(94) = 5758  that follow  a(92) =1438  are the only ones on rays with slope 32 resp. 64, for  n < 10^4.

There are other rays with slope < 1, I'd call these "prime rays", because all primes (and only primes) lie on these rays.

Most primes lie on the main prime ray with slope slightly less than 1/2 :
a(n) / n ~ 1/2 - 1 / 202  around  n ~ 1e4.
The secondary prime ray has slope ~ 2/3 : a(n) / n ~ 2/3 - 1 / 165  around n ~ 1e4 Then there's a third prime ray with slope ~ 1/3, but only 2 + 5 of all  a(n = 8000 .. 10^4) are on that ray.
Then there are some more prime rays, for example one with slope a(n) / n = 4/5, but again, even fewer values (only 4 among a(6000 .. 10^4)), are on these.

- Maximilian

(PARI) for my records and those intersted \r /tmp/b2v.gp

%398 = 19882
%399 = 1.9951831409934771700953336678374310085
%401 = 9968
%402 = 0.99869752529806632601943693016731790402
 [x|x<-A13,x>1e4 && !isprime(x/2)]  \\ all values > 1e4 are even semiprimes
%449 = []
 [n | n<-[1..10^4], A13[n] > 0.9*n && isprime(A13[n]) ]
%474 = [1423]  \\ the only n with prime a(n) and  a(n)/n > 0.9

 [n | n<-[6000..10^4], A13[n] > 2/3*n && A13[n] < 0.95*n  ]
%498 = [5011, 5059, 5683, 6619]
%499 = [0.79050323394857233002050796655623915444,
%500 = [105, 102, 118, 116]

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