collatz conjecture solved

I want to generate a sequence according to Collatz Conjecture "Given an integer n, if it is even, divide n by 2, (n/2) , otherwise if it is odd, triple n and add it to 1 (3n +1). In regards to testing, it may be the case that some Conjectures can never be formally proven. The net effect being that there is a higher probability of a divide occuring than a multiply, resulting in a trend towards 1. Earlier this year one of the top mathematicians in the world dared to confront the problem — and came away with one of the most significant results on the Collatz conjecture in decades. Applying it to 8 we get 4. Take any positive integer: if the number is even, divide it by two; if the number is odd, triple it and add one (for example, if this operation is performed on 26, the result is 13; if it is performed on 5, the result is 16). If the integer is odd, multiply it by 3 and add 1 to the result (3a1+ 1) to get the next number in the sequence. The Python Code to solve Collatz Conjecture example. Collatz Conjecture (3x+1 problem) states any natural number x will return to 1 after 3 x+1 computation (when x is odd) and x/2 computation (when x is even). Repeat above two steps with new value. If n is odd, multiply n by 3 and add 1 to get 3n + 1. Details in link: The first portion of the Conjecture prevents the ability of the algorithm terminating with an odd number and the second portion does the same except for the pattern 2^x. Are we one step away from a complete solution? If odd multiply by 3 and add one. Tao’s breakthrough post is titled “Almost All Collatz Orbits Attain Almost Bounded Values.” Let’s break that down slightly. If n is even, divide n by 2. If n is odd, multiply n by 3 and add 1 to get 3n + 1. Now that’s odd, so we multiply 5 by 3 and then add 1, landing us on 16. Yet more obvious: If N is odd, N + 1 is even. Then we get 2 and then we get 1. (If negative numbers are included, there are four known cycles (excluding the trivial … Although the problem on which the conjecture is built is remarkably simple to explain and understand, the nature of the conjecture and the be- havior of this dynamical system makes proving or disproving the conjecture exceedingly difficult. It’s a siren song, they say: Fall under its trance and you may never do meaningful work again. The cartoon is accurate but let's make the conjecture clear: Pick a number, a positive integer. Today is my anniversary on WordPress, so to celebrate I decided to solve the Collatz Conjecture. Despite this small step towards the solution to the problem, almost all mathematicians agree that the complete answer to … Change ), You are commenting using your Google account. the Collatz conjecture) is solved if we prove that the OCS of any odd number is finite. In this paper, we propose a new approach for possibly proving Collatz Conjecture (CC). On Sept 8th Terence Tao uploaded a paper which stated that the Collatz Conjecture was “almost true” for “almost all numbers”. the Collatz conjecture) is solved if we prove that the OCS of any odd number is finite. The Collatz conjecture concerns what happens when we take any positive integer n and apply the following algorithm: The conjecture states that when this algorithm is continually applied all positive integers will eventually reach 1. In this paper, we propose a new approach for possibly proving Collatz Conjecture (CC). Although the problem on which the conjecture is built is remarkably simple to explain and understand, the nature of the conjecture and the be-havior of this dynamical system makes proving or disproving the conjecture … [1] It is also known as the 3n + 1 problem, the 3n + 1 conjecture, the Ulam conjecture (after Stanisław Ulam), Kakutani’s problem (after Shizuo Kakutani), the Thwaites conjecture (after Sir Bryan Thwaites), Hasse’s algorithm (after Helmut Hasse), or the Syracuse problem. In a nutshell, an elliptic curve is a special kind of function. This still wouldn’t be a formal proof. It is an open question if all formal proofs can be validated in a reasonable timeframe. So you could call this a very powerful new branch of math. For example, 10, 5,16, 8, 4, 2, 1. Therefore, it is an open question if all problems can be formally proved. Collatz Orbits are just the little sequences you get with the process we just did. Since (N + 1) is odd, 3(N + 1) + 1 is even. September 6, 2015 17:31 1 INTRODUCTION We just write OCS if we mean an arbitrary odd Collatz sequence or if the seed is known and in plural form we write OCS’s.Obviously 3n + 1 (i.e. • The OCS of a number x is cyclic in the same way that a Collatz sequence is cyclic, i.e. Thwaites (1996) has offered a £1000 reward for resolving the conjecture . Collatz Conjecture is a numbers problem that is even older and has been giving even the brightest minds the run for their money. Hn is the n … Take any natural number. The Collatz conjecture, also known as the 3n+1 conjecture and other names), deals with the following operation to produce a sequence of numbers. The way I look at it is that what you are describing is a conjecture, which in math is a statement that is true in all tested cases but can’t be logically proven yet. Fill in your details below or click an icon to log in: You are commenting using your WordPress.com account. Collatz Conjecture Calculator: Enter Natural Number for Collatz Conjecture (1,2,...,∞): Collatz Conjecture Video A refresher on the Collatz Conjecture: It's all about that function f(n), shown above, which takes even numbers and cuts them in half, while odd numbers get tripled and then added to 1. The conjecture is that if you apply f(n) to an integer enough times in a row it will eventually reach a value of 1 at some point. Can /sci/ solve the issue of the Collatz Conjecture? Answered. The first step is to define a new function called “Collatz”. Given any positive integer n, define . Collatz Conjecture . long-awaited answer to a decades-old math problem, Almost All Collatz Orbits Attain Almost Bounded Values, impossible math problems were eventually solved, Physicist Solves 127-Year-Old Wave Riddle, Riddle Solution: The Gold Chain Math Problem, Solution to Riddle of the Week: The Doodle Problem, Mathematician Solves Old, Famous Knot Problem, Riddle of the Week #1: The Farmer's Dilemma, Riddle of the Week #10: Einstein's Riddle. The start of a bias. The problem with the conjecture is that it is unproven but in practice for every number tested it results in the number 1 (eventually). [solved] Collatz Conjecture in Spreadsheet. Change ), You are commenting using your Facebook account. For example, 10, 5,16, 8, 4, 2, 1. Carnegie Mellon University computer scientists and mathematicians have resolved the last, stubborn piece of Keller's conjecture, a geometry problem that scientists have puzzled over for … It doesn’t actually matter what your function is called, but choosing a name that is logical is a good habit to keep. Using the spreadsheet I enter 27 in cell A1, and in cell A2 I enter For all we know it will take decades, and completely new branches of math, to finally be put to rest. This week, we’ve celebrated the long-awaited answer to a decades-old math problem, and now we’re one step closer to an even older numbers puzzle that has stumped the world’s brightest minds. Apply the same rules to the new number. If you could execute the program for all whole numbers, then you could validate the correctness of the argument and make a claim of a formal proof. [7], https://en.wikipedia.org/wiki/Collatz_conjecture. The Collatz Conjecture or 3x+1 problem can be summarized as follows: Take any positive integer n. If n is even, divide n by 2 to get n / 2. If n is odd, multiply n by 3 and add 1 to get 3n + 1. … Since 3 is odd, we get the next term in th… More info and links in full description. Can /sci/ solve the issue of the Collatz Conjecture? ‍♂️. If it’s odd, multiply it by 3 and add 1. Not a bad effort. One of the best things about Tao is that he really delivers on content, and openly shares it with the world. Thwaites (1996) has offered a £1000 reward for resolving the conjecture . Well, kind of. Repeat the process indefinitely. Given a positive number, n, if n is even then the next number is n divided by 2. They could exist, but their frequency approaches 0 as you go farther down the number line. The Collatz conjecture, also known as conjecture , conjecture of Ulam or problem of Syracuse, is a conjecture of number theory established by Lothar Collatz in … Collatz Conjecture . It was solved by Sir Andrew Wiles, using Elliptic Curves. Start with a positive number n and repeatedly apply these simple rules: If n = 1, stop. The Collatz conjecture is a conjecture in mathematics that concerns a sequence defined as follows: start with any positive integer n. Then each term is obtained from the previous term as follows: if the previous term is even, the next term is one half of the previous term. Proposed in 1937 by German mathematician Lothar Collatz, the Collatz Conjecture is fairly easy to describe, so here we go. Posted on 10 September 2019 by John. Now the last obvious bit: If N is even, N + 1 is odd. fnews, the problem isn't fully solved. (You were warned!) His blog is like a modern-day da Vinci’s notebook. If you try it you will discover that you eventually reach a result of 1. When I observed the first part of the Conjecture, I noted that it was basically to push an odd result to an even one. As someone from an applied math background, I would like to have formal proofs for a restricted domain as this has practical applications. I happened to spot this on Slashdot earlier today and, to be honest, it was the first time I saw it. Popular Mechanics participates in various affiliate marketing programs, which means we may get paid commissions on editorially chosen products purchased through our links to retailer sites. Windows applications require the Microsoft Visual C++ Redistributable for Visual Studio 2017 . In solving this, I noted that it just comes down to what pattern you spot, rather than any genuine effort or capability. We propose Reduced Collatz Conjecture (RCC)—any natural number x will return to an integer that is less than x. UNCRACKABLE? Gear-obsessed editors choose every product we review. Collatz Conjecture (3x+1 problem) states any natural number x will return to 1 after 3 x+1 computation (when x is odd) and x/2 computation (when x is even). The Collatz conjecture, also known as the 3n+1 problem, asks whether the following function terminates for all positive integer arguments n. def collatz(n): if n == 1: return 1 elif n % 2 == 0: return collatz(n/2) else: return collatz(3*n+1) There’s a deep meaning to how rare we’re talking here, but it’s still very different from nonexistent. Within a few seconds, I solved it. It could be answered by looking at the properties of another, additive-type function that produces for every Collatz sequence an odd subset of the same numbers, in the same order, between n and 1. Where n is a positive integer. If we restrict the domain to 3-10000, we could certainly claim that the program is a formal proof for that restricted domain.”. People become obsessed with it and it really is impossible,” said Jeffrey Lagarias, a mathematician at the University of Michigan and an expert on the Collatz conjecture. Windows applications require the Microsoft Visual C++ Redistributable for Visual Studio 2017 . Is there a difference between testing the underlying assumptions and testing of an output? And while the story of Tao’s breakthrough is good news, the problem isn’t fully solved. A refresher on the Collatz Conjecture: It's all about that function f(n), shown above, which takes even numbers and cuts them in half, while odd numbers get tripled and then added to 1. We offer a humble, yet seemingly paltry, contribution to this endeavor by proving the extremely important Collatz Conjecture with many applications (see section 5), which states: 1.1 Collatz Conjecture . As such, theoretical mathematicians will argue that the Collatz Conjecture has been isolated further to whether the formula will discover the pattern 2^x in execution. The Collatz conjecture is a conjecture in mathematics that concerns a sequence defined as follows: start with any positive integer n. Then each term is obtained from the previous term as follows: if the previous term is even, the next term is one half of the previous term. If x+y=z then I can prove that z-y=x. A test is not necessary in a formal proof. Hopefully that makes sense, sorry I’m so bad at explaining it. 2. On September 8, Terence Tao posted a proof showing that — at the very least — the Collatz conjecture is “almost” true for “almost” all numbers. Carnegie Mellon University computer scientists and mathematicians have resolved the last, stubborn piece of Keller's conjecture, a geometry … [2][4] The sequence of numbers involved is sometimes referred to as the hailstone sequence or hailstone numbers (because the values are usually subject to multiple descents and ascents like hailstones in a cloud),[5][6] or as wondrous numbers. Now you have a new number. •The OCS of a numberxiscyclicin the same way that a Collatz sequence is cyclic, i.e. The conjecture is that no matter what value of n, the sequence will always reach 1. Equation: σ (n) ≤ Hn +ln (Hn)eHn. We propose Reduced Collatz Conjecture (RCC)—any natural number x will return to an integer that is less than x. So if you’re looking for a counterexample, you can start around 300 quintillion. Now 16 is even, so we cut it in half to get 8. Take any positive integer: if the number is even, divide it by two; if the number is odd, triple it and add one (for example, if this operation is performed on 26, the result is 13; if it is performed on 5, the result is 16). This article deals with a different class of formal proof. Note that the answer would be false for negative numbers. The code is functional and extensive testing has yet to reveal an error. A formal proof shows *why* the conjecture is always true using *logic* not testing. Take any natural number. Today is my anniversary on WordPress, so to celebrate I decided to solve the Collatz Conjecture. jonbenedick shared this question 5 years ago . While Tao’s result is not a full proof of the conjecture, it is a … If the previous term is odd, the next term is 3 times the previous term plus 1. Details in link: just check if n is a positive integer or not. This function will accept a number. Solved: The Collatz Conjecture. 3. (1) always returns to 1 for positive . In the above code, the best we can conclude is that the brute force search will discover the pattern 2^x in all tested cases. No testing needed. The Collatz conjecture states that the orbit of every number under f eventually reaches 1. Well, even Tao says no. The goal remains to prove they don’t exist whatsoever. The Collatz Conjecture: A Brief Overview Matthew Hammett The Collatz conjecture is an elusive problem in mathematics regarding the oneness of natural numbers when run through a specific function based on being odd or even, specifically stating that regardless of … The conjecture is about what happens as you keep repeating the process…, …But Collatz predicted that’s not the case. The Collatz Conjecture: A Brief Overview Matthew Hammett The Collatz conjecture is an elusive problem in mathematics regarding the oneness of natural numbers when run through a specific function based on being odd or even, specifically stating that regardless of the initial number the series will eventually reach the number 1. Thanks for the reply. The Great Courses Plus (free trial): http://ow.ly/RqOr309wT7v This video features Alex Bellos. From a practical viewpoint as a programmer, describing the problem as solved is potentially satisfactory. The cartoon is accurate but let's make the conjecture clear: Pick a number, a positive integer. Let's play a little game. ( Log Out /  The problem with the conjecture is that it is unproven but in practice for every number tested it results in the number 1 (eventually). At 24, he became the youngest math professor at UCLA⁠—ever. And when, 3x+1is an even number, we can successfully halve it according to first step of the function defined in the conjecture. But many mathematicians, including the one responsible for this newest breakthrough, think a complete answer to the 82-year-old riddle is still far away. I’m well aware of what constitutes a formal proof. math. The Collatz conjecture, also known as conjecture , conjecture of Ulam or problem of Syracuse, is a conjecture of number theory established by Lothar Collatz … Then one form of Collatz problem asks if iterating. So for practical purposes you can usually assume that a conjecture is true because it hasn’t been proven false. So what does it mean here? The suggestion is to leverage the testing process from computer programming and lower the standard of formal proof from all cases, to all testable cases. It has been speculated that we require new mathematical tools to prove this Conjecture, but it does seem increasingly likely that we need to review practices. If odd multiply by 3 and add one. If it’s even, divide it by 2. I tested this latter assumption with some code: This code proved that there were indeed more even numbers in a given range than odd. The Riemann Hypothesis. From a theoretical mathematics perspective, the classical viewpoint would be that the above is not a proof, as a proof needs to hold for all cases. How we test gear. Why hasn't the Collatz Conjecture been solved yet? So, the Collatz conjecture seems to say that there is some sort of abstract quantity like 'energy' which cannot be arbitrarily increased by adding 1. Think of the program as a logical argument that the indicated solution in the article is correct. Since 3x+1 is an even number for any odd x, we can replace any odd number by an even number which equals to 3x+1. You may be able to find more information about this and similar content at piano.io, This TikTok Star Uses Math to Guess Your Height, We Already Know How to Build a Time Machine, No One Can Figure Out How to Cut Christmas Cookies, The Geometry Behind This Viral Gift-Wrapping Trick, Mathematician Makes Quadratic Equations Easier. For example, consider starting with the integer 3. Air Force's Secret New Fighter Comes With R2-D2, Mathematician Solves the Infamous Goat Problem, Three Asteroids to Fly Past Earth on Christmas Day, In 1944, POWs Got a Great X-Mas Gift—An Escape Map, How to Solve the Infuriating Viral Math Problem, College Board Gets Complex SAT Math Problem Wrong, This content is created and maintained by a third party, and imported onto this page to help users provide their email addresses. So this week, Tao takes us to the Collatz Conjecture. A program to calculate the Collatz Conjecture with frequency counts. there exists a numbery ∈2N + 1 such thatyoccurs twice in the OCS. The Collatz Conjecture or 3x+1 problem can be summarized as follows: Take any positive integer n. If n is even, divide n by 2 to get n / 2. fnews, the problem isn't fully solved. We may earn commission if you buy from a link. f ( n) = { n + n + 1 2, if n + 1 ≡ 0 mod 4 n − n − 1 4, if n − 1 ≡ 0 mod 8 n − n + 1 2 2, otherwise. The big detail in Tao’s proclamation is that first “Almost.” That word is the last barrier to a full solution, and it takes different meanings in different math contexts. Only 36 Percent of People Can Pass This Logic Test, Everyone's Trying This Annoying Math Challenge, How to Solve the SAT Question Everyone Gets Wrong. So, now that we know its counterexamples are rarer than ever, where does that leave the problem? n is ≥ 4. This function will accept a number. This raised the issue of a formal proof being potentially an unrealistic goal because of the validation issue, rather than actual incorrectness. The Collatz Conjecture or 3x+1 problem can be summarized as follows: Take any positive integer n. If n is even, divide n by 2 to get n / 2. Abstract. Since this is unfeasible, the problem remains a Conjecture. A proof is something that has been logically proven. In essence, Tao’s results says that any counterexamples to the Collatz Conjecture are going to be incredibly rare. TOPIC. There is … There is a rule, or function, which we apply to that number, to get the next number. Name a subject in advanced math, and he’s written about it. The Collatz conjecture, also known as the 3n+1 problem, asks whether the following function terminates for all positive integer arguments n. Ifnis odd, then the next number is 3n+1. For those that don’t know the Conjecture, here are the basics: The conjecture is named after Lothar Collatz, who introduced the idea in 1937, two years after receiving his doctorate. The Collatz Conjecture - Numberphile - YouTube I’m using the Collatz Conjecture as an example. Given a positive number, n, if n is even then the next number is n divided by 2. If you try it you will discover that you eventually reach a result of 1. It’s describing how rare the counterexamples to the Collatz Conjecture are, if they exist at all. Change ), Prince Andrew: The Fake Virginia Roberts Photo. The Collatz conjecture remains today unsolved; as it has been for over 60 years. In a practical sense, probably not, its just that one may get more testing than the other. I want to generate a sequence according to Collatz Conjecture "Given an integer n, if it is even, divide n by 2, (n/2) , otherwise if it is odd, triple n and add it to 1 (3n +1). Repeat above two steps with new value. Collatz cycles can be shown to imply a difficult result in number theory: Theorem: The gap between powers of 2 and powers of 3 goes to infinity. He conjectured that if you start with a positive whole number and run this process long enough, all starting values will lead to 1. The conjecture is named after Lothar Collatz, who introduced t Terence Tao is one of the greatest mathematicians of our time. f(n) = 3n+1 if n is odd and f(n)=n/2 if n is even . In the comments to the blog post, he says, “one usually cannot rigorously convert positive average case results to positive worst case results, and when the worst case result is eventually proved, it is often by a quite different set of techniques.” In other words, this cool new method may give us a near-solution, but the full solution might take an entirely different approach. One where it is unfeasible to validate correctness in a reasonable timeframe. The Python Code to solve Collatz Conjecture example. So the Collatz Orbit of 10 is (10, 5, 16, 8, 4, 2, 1, 4, 2, 1, …). This article describes the Collatz Conjecture as solved, but does it amount to a formal proof? factoring out a power of 2 has a small effect on the factorization (in that it doesn't change the other prime powers in the factorization). Just logic. The Collatz conjecture is for computer science what until recently Fermat’s last theorem was for mathematics: a famous unsolved problem that is very simple to state. Then one form of Collatz problem asks if iterating. The Collatz conjecture is quite possibly the simplest unsolved problem in mathematics — which is exactly what makes it so treacherously alluring. By the induction hypothesis, the Collatz Conjecture holds for N + 1 when N + 1 = 2k. Let, f(x)=x/2 if x is even and g(x)=3x+1 if x is odd. It states that if n is a positive then somehow it will reaches to 1 after a certain amount of time. Today's High Steps. Repeat for the each term. Create a sequence, or list, of numbers using the following rules: 1. The Collatz conjecture, also known as the 3n+1 conjecture and other names), deals with the following operation to produce a sequence of numbers. It’s even, so the rule says to divide by 2, taking us to 5. On Sept 8th Terence Tao uploaded a paper which stated that the Collatz Conjecture was “almost true” for “almost all numbers”. Not a bad effort. Since half of 4 is 2, half of 2 is 1, and 3*1+1 is 4, Collatz Orbits cycle through 4, 2, and 1 forever. If even divide by 2. For example, let’s use 10. Once a pattern of 2^x is found (i.e. It’s definitely true for all numbers with less than 19 digits, so that covers whatever you probably had in mind. But even if computers check up to 100 or 1,000 digits, that’s far from a proof for all natural numbers. Since it's odd, the Collatz function returns 16. That’s the Collatz Conjecture. At age 21, he got his Ph.D. at Princeton. That is, a proof is only a proof because the underlying assumptions have been subjected to extensive testing. Not some form of intrinsic truth devoid of practical considerations. In 1937, Lothar Collatz asked whether this procedure always stops for every positive starting value of n. If Gerhard Opfer is correct, we can finally say that indeed it … If that is the case, why would it matter at what point the testing was done? So mathematicians will use Tao’s newest innovations to solve (or nearly solve) other major problems, but it looks like the Collatz Conjecture itself still remains unfinished. “Think of the program as a logical argument that the indicated solution in the article is correct. Mathematicans are complaining that some proofs are so large and so specialised that they are unable to confirm correctness. The Collatz Conjecture project makes use of the parity sequence optimization and runs on Linux, Windows, and OS X and can utilize CPUs as well as AMD, nVidia, and Intel graphics cards. I have been watching the debate on this online and it is beginning to centre around whether or not a proof is, ultimately, of similar quality to the code provided. It doesn’t actually matter what your function is called, but choosing a name that is logical is a good habit to keep. And in 2006 he won the Fields Medal, known as the Nobel Prize of math, at the age of 31. Obviously 3n+ 1 (i.e. The technical term in this case is logarithmic density. ( Log Out /  And while no one has proved the conjecture, it has been verified for every number less than 2 68. If n is odd, multiply n by 3 and add 1. Experienced mathematicians warn up-and-comers to stay away from the Collatz conjecture. Even again, so halving gets us 4. Perform this operation repeatedly, beginning with … “This is a really dangerous problem. The conjecture states that no matter which number you start with, you will … Change ), You are commenting using your Twitter account. The Collatz Conjecture has been solved as a brute force search for the pattern 2^x and it holds for all numbers. The idea is to use Collatz Conjecture. Now 4 is even, so we take half, getting 2, which is even, and cuts in half to 1. The Collatz Conjecture project makes use of the parity sequence optimization and runs on Linux, Windows, and OS X and can utilize CPUs as well as AMD, nVidia, and Intel graphics cards. In this case, the OCS is obviously also infinite. Gerhard Opfer has posted a paper that claims to resolve the famous Collatz conjecture. 32-23 = 9-8 = 1; 25-33 = 32-27 = 5; 28-35 = 256-243 = 13; 37-211= 2187-2048 = 139; … Basically, if a power of 2 and power of 3 are too close together, they can be used to create a Collatz cycle. And once you hit 1, the rules of the Collatz conjecture confine you to a loop: 1, 4, 2, 1, 4, 2, 1, on and on forever.”, https://www.quantamagazine.org/mathematician-terence-tao-and-the-collatz-conjecture-20191211/. And when, 3x+1is an even number, we can successfully halve it according to first step of the function defined in the conjecture. The problem I always had is coming face to face with a real-world problem that could be solved with math, being able to recognize it could be solved with math, knowing which math concept(s) are involved, and then and only then, remembering how to solve that type of problem. Now, applying the Collatz function to 16, we get 8. The conjecture is that no matter what value of n, the sequence will always reach 1. If we restrict the domain to 3-10000, we could certainly claim that the program is a formal proof for that restricted domain. Proposed in 1937 by German mathematician Lothar Collatz, the Collatz Conjecture is fairly easy to describe, so here we go. Write a C program using fork() system call that generates this sequence in the child process. Using the spreadsheet I enter 27 in cell A1, and in cell A2 I enter Collatz Conjecture Calculator: Enter Natural Number for Collatz Conjecture (1,2,...,∞): Collatz Conjecture Video If the previous term is odd, the next term is 3 times the previous term plus 1. That is, it is still a Conjecture. Its probably not true of all efforts in the field, but it would be interesting to learn how many had a similar experience. Let, f(x)=x/2 if x is even and g(x)=3x+1 if x is odd. How Would You Solve This Hard Letter Math Problem? (If negative numbers are included, there are four known cycles (excluding the trivial 0 cycle): (4, 2, 1), (, ), (, , … This article is highlighting that the process of formal proof validation is extremely difficult. 2, 4, 8, 16, 32, 64, 128, etc), it will then reduce to 1 and repeat the pattern 1, 4, 2, 1, 4, 2, 1, etc. If the integer is even, divide it by 2 to get the next number in the sequence (a1 / 2). ( Log Out /  Take any natural number, apply f, then apply f again and again.

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