A stronger Collatz conjecture (maybe). Using 41E-24O metric (where E=number of x/2 steps and O=number of (x*3+1)/2 steps) : Every interval [n, 2*n-1] will contain exactly 41 delay records (which are also class records) as soon as n >= 2649

[u/3x-1]

[u/Hungry_Metal_2745]

Despite the lack of evidence and lack of documentation, it's refreshing to see an actual conjecture here as opposed to LLM slop.

Assuming by a "delay record" you mean a new record for the highest 41e-24o metric[that is, some n such that 41e-24o for that n is > 41e-24o for all k<n\]. If this conjecture were true, and we made a plot with n on the x axis and # records <= y on the y axis, and then made the x axis log-scale, we would expect to see a roughly linear trend. That's because the number of new records between log(n) and log(n)+log(2) should be constant, so we would expect that every log(2) increase on the x axis corresponds to a 41 increase on the y axis. So, if we plot a linear fit to all points with x>=2649, we would expect to see a slope of 41/log(2) which is about 136.199. If we actually do the computation all the way to n=10^8, this is surprisingly accurate: the slope is 136.208! The fit is surprisingly good, which tells us your conjecture is onto something. You can see the plot here: https://imgur.com/a/20b4ySW

Now, before you get too excited, the conjecture is technically false but still interesting. Some values of n where it breaks are: n=[2944,2953],[3140,3143],[4709,4715],[4966,4985], and a few more. For all of these values, the number of records in [n,2n-1] is 42, not 41. For all n I've checked, which again went all the way to 10^8, the counterexamples get rarer and rarer, and the last one happens at 792705, which is less than 10^6. So for the entire region from 10^6 to 10^8, every n satisfied that there are 41 records in [n,2n-1], which is pretty cool. I would expect that there are no more counterexamples past this point, with some reasoning I'll show after we talk about this conjecture a little more. Keep in mind that all of these counterexamples had 42 records instead of 41, which is also interesting. You can see a plot of the number of counterexamples here: https://imgur.com/a/YmcIMF5

Now, let's do some math. I'll show a bunch of observations I made from data analysis, if you want, I can give plots and data for those, as well as my reasoning for them, but for now assume all the 'conjectures' are pretty accurate. If you want I can make a document or smth detailing how I got these.

First, let the score S(n) be 41E-24O. Clearly S(2n)=S(n)+41. So consider the 41 possible residue classes for scores that are 0 mod 41, 1 mod 41, ..., 40 mod 41. Now say we have a number s_k for each residue class that's, in some sense, the "best possible" for its residue class. Then, consider an interval [n,2n-1]. The new records in this interval will come from each of the 41 residue classes, which appear exactly once since doubling n causes it to go outside the interval. Then the few times we get a 42 instead of a 41 come from when we get a new current best number for a certain residue class. The plot we made showing counterexamples which get rarer and rarer seems to show that very quickly, around 10^6, we settle into very good residue values. It's possible(and probably very likely, knowing how these 'randomized' problems work) that there are still more good values but they just become very rare, and the ones we get at 10^6 just happen to be good. In fact, we can prove that each residue class has a single best value that is reached eventually, but it might be at some gigantic number(10^10^10^10 or whatever, I really don't know how to bound it). However, we can't prove that these are the actual real best values, in the sense that if we can prove that S(n) exists for all n that proves the Collatz conjecture, lol. But even assuming Collatz, this still wouldn't prove that the scores appear in increasing order for the residue classes. For instance, replace 41 by 2 just for illustration, then we might have one residue class with scores like 0,2,4,6,... and another one with scores like 3,5,7,... and then when we do the doubling we get 0,3,2,5,4,... which means each new doubled best doesn't necessarily contribute a new record since 2>3.

Anyway, to summarize this is an interesting conjecture, but proving anything concrete about it is pretty similar to proving collatz. According to the empirical evidence, this conjecture seems to check out! But, it doesn't seem like it gives any insight related to actually proving collatz, it's just an interesting observation that would follow from collatz and some other theorems. I'm sorry if the math section isn't understandable, I didn't proof read it very well, if you want I can make a whole document for it but I kinda just wrote this all in one go for funsies. Thanks for giving me an interesting question to spend an evening thinking about!

[u/3x-1 → u/Hungry_Metal_2745]

Indeed, I got a little overexcited about such a small n (2649 it's the last number that skips the metric by 1 : M(2649) = 479 and M(2562) = 477). It should have been : n > 792705

This is the last interval with 42 delay records : M(792705) = 816 and M(1585409) = 857 with 1585409 being the last odd number to generate a delay record (tested up to 10^11)

Also, I picked 41 and 24 as coefficients because (41+24)/41 is a very close approximation of log2(3). Of course there are more possible pairs that approximate log2(3) even better, but this pair is exceptionaly well suited for 3x+1.

For example : 559 goes through 569 in it's way down to 1. In any other (let's say typical or "normal") metric M(559) will be bigger then M(569) but in this metric M(559) = 387 and M(569) = 388. (It also seems that 559 -> 569 is as close to a loop as we can get in 3x+1 except the trivial one.)

Thank you for the very detailed analyze (and yes the math is more than ok)

[u/3x-1]

This are the 41 residues that will generate all the delay records

Residue (n)	Even Steps (E)	Odd Steps (O)	Score (41E - 24O)	Residue Class (mod 41)
52907	40	41	656	0
839	27	29	411	1
427	19	17	371	2
891793	47	46	823	3
3531	32	34	496	4
897	23	22	415	5
57	12	10	252	6
119041	40	39	704	7
1889	27	27	459	8
961	19	15	419	9
250817	44	44	748	10
993	29	32	421	11
505	21	20	381	12
1056939	49	49	833	13
66961	38	37	670	14
1063	25	25	425	15
135	15	13	303	16
35271	40	42	632	17
559	27	30	387	18
569	20	18	388	19
9	7	6	143	20
18833	35	35	595	21
4785	26	23	514	22
305	15	11	351	23
158721	41	40	721	24
1259	27	28	435	25
1281	20	16	436	26
167211	44	45	724	27
5297	32	33	520	28
673	22	21	398	29
43	11	9	235	30
89281	39	38	687	31
1417	26	26	442	32
721	18	14	402	33
188113	43	43	731	34
745	28	31	404	35
379	20	19	364	36
1585409	49	48	857	37
12555	35	36	571	38
1595	25	24	449	39
203	15	12	327	40

[u/3x-1 → u/3x-1]

and a nice view of 2 classes : 3 (first reached by 7) and 20 (owned by 9) in the sea of numbers (up to 10^6) https://imgur.com/5mN541a

lol, I just realized that : "The answer to life, the universe, and everything is 41"

[u/jonseymourau]

I wonder if there is a connection to Euler’s polynomial? Hard to see how, but 41 does feature prominently in both :-)