[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
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.)
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
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.
(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 =  \\ 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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