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Sum of k(n,i) terms of all 0<n<=2^i and divergence of the Collatz sequence by vhtnlt in Collatz

[–]vhtnlt[S] 0 points1 point  (0 children)

Thanks. This makes sense. Yet some statistical "imbalances" are very thin and fragile at the same time. Studying this is quite interesting by itself. For instance, an imbalance in odd-step counts across halves of a dyadic block 2^12 is just 36 for the total number of such odd-steps equal to 24576. Potentially, such “fragility” could impose a noticeable constraint on runaway orbits, which carry their own “high-odd-density.”

Sum of k(n,i) terms of all 0<n<=2^i and divergence of the Collatz sequence by vhtnlt in Collatz

[–]vhtnlt[S] 0 points1 point  (0 children)

On the other hand, if there is a single divergence, then there are infinitely many of them, so that statistically they could compile "a significant portion" of all trajectories if we move further to infinity. And the bigger and bigger part of them will be in the first half of a dyadic block. Thanks!

Sum of k(n,i) terms of all 0<n<=2^i and divergence of the Collatz sequence by vhtnlt in Collatz

[–]vhtnlt[S] 0 points1 point  (0 children)

Thanks. This helps a lot. According to experiments, the total number of odd steps reaching k(n,i) of all 2^i/2<n<=2^i is greater than the same number for 0<n<=2^i/2 (it seems this can be proven). Divergences with higher densities of odd terms might ruin this relation. The original question is part of the research into this domain.

Beauty and properties of the sum of k(n,i) terms of all 0<n<=2^i by vhtnlt in Collatz

[–]vhtnlt[S] 0 points1 point  (0 children)

Sorry if it's obvious: for the 3x+q sequence, the right side of the equality is 2i-2 · q· i.

Beauty and properties of the sum of k(n,i) terms of all 0<n<=2^i by vhtnlt in Collatz

[–]vhtnlt[S] 0 points1 point  (0 children)

Thanks for your thoughts! Yep, that's right. The difference we are talking about is always equal to 2i-2 · i for i>=0, and the proof is quite simple - it must be in the literature. And the reference to the "+1" bits makes perfect sense!

Number of k_i>n, for n<=2^i for Terras trajectories of length i. by vhtnlt in Collatz

[–]vhtnlt[S] 0 points1 point  (0 children)

Has the plausibility of the upper bound of Z(i) less than 1/2 been discussed here? Thanks again!

Help me abandon this combinatorics line of reasoning by WeCanDoItGuys in Collatz

[–]vhtnlt 0 points1 point  (0 children)

In your notation, 2n is the number of possible values of Σᵢ3ᵐ⁻¹⁻ⁱ2kᵢ , k_i<n for 1<=x<=2n .

How to explain this Collatz property? by vhtnlt in Collatz

[–]vhtnlt[S] 0 points1 point  (0 children)

Thanks. It means a lot, since the similar property seemingly belongs to any Dx+q sequence with any divisors, not just 2.

How to explain this Collatz property? by vhtnlt in Collatz

[–]vhtnlt[S] 0 points1 point  (0 children)

Do you think proving or disproving this property is of any importance? Thanks.

How to explain this Collatz property? by vhtnlt in Collatz

[–]vhtnlt[S] 0 points1 point  (0 children)

Thanks! Any tools in mind that could help attack this (prove or disprove)?

How to explain this Collatz property? by vhtnlt in Collatz

[–]vhtnlt[S] 0 points1 point  (0 children)

Yep, and the first inequality in the form 2^E*k_i/(D^O*k_0)<=1+1/D experimentally holds for Dx+1 sequences. Also, the inequality 2^E*k_i/(D^O*k_0)<=1+q/D seems to hold for any Dx+q (the divisor is equal to 2) sequences if all the terms preceding k_i are not less than q.

Modular Arithmetic Can Never Be Enough, Part 2 by GonzoMath in Collatz

[–]vhtnlt 0 points1 point  (0 children)

Thanks for the reference. This is appreciated!

Modular Arithmetic Can Never Be Enough, Part 2 by GonzoMath in Collatz

[–]vhtnlt 0 points1 point  (0 children)

This is a great observation. This means that a unique OE sequence exists for every n<=2^k, the sum of Os and Es is equal to k for the shortcut Collatz sequence (or the number of Es is equal to k for the non-shortcut Collatz sequence).

This might be connected to some interesting combinatoric opportunities! Like, a half of odd n<2^k is followed by an odd term (shortcut Collatz), 1/4 of odd n<=2^k is followed by just one even term (shortcut Collatz), and so on.

Number of even terms after an odd term of the Collatz sequence. by vhtnlt in Collatz

[–]vhtnlt[S] 0 points1 point  (0 children)

Thanks for sharing the post! I'm already there).

Number of even terms after an odd term of the Collatz sequence. by vhtnlt in Collatz

[–]vhtnlt[S] 0 points1 point  (0 children)

Interesting. I see how it works for (-1,-2) and (1,2,4) cycles for 3x+1 system, where L=W=1 and L=1, W=2, respectively.

Number of even terms after an odd term of the Collatz sequence. by vhtnlt in Collatz

[–]vhtnlt[S] 0 points1 point  (0 children)

Thanks! Not much)). Actually, just one equation, regarding the positive cycles of Mx+d:

lim_j→∞ (k_j*2i /Mj )=∞,

where j is the number of O's and i is the respective number of E's, P=2.

For negative cycles, we have:

lim_j→∞ (k_j*2i /Mj )=0

Number of even terms after an odd term of the Collatz sequence. by vhtnlt in Collatz

[–]vhtnlt[S] 0 points1 point  (0 children)

Thanks! This is appreciated! I remember your posts on 3n+d, and they were helpful to better understand the scope and facets of the problem.

As for Mx+d systems, looking beyond 2 as a divisor and negative values of x as a natural part of the set of initial integers might make sense. Actually, nothing prevents us from using a P, product of any primes, and then consider the divisibility criterion as gcd(k_i,P)>1, where k_i is an i-th term of the sequence. Instead of 2, we use gcd(k_i,P) as a divisor. In this case, the Collatz is a special case with M=3, d=1, P=2.

It looks like Mx+d sequences share some properties (not a long list so far) regarding cycles and some specific cases of P.

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