A381246 Contribution to the OEIS (supporting a juggler sequence)

 A381246

Largest value in trajectory of n under the juggler map of A380891.

0

1, 2, 4, 4, 8, 6, 30, 8, 18, 10, 24, 12, 30, 14, 36, 16, 150, 18, 50, 20, 1320, 22, 43366048, 24, 41678, 26, 350, 28, 41678, 30, 234421146, 32, 2438232, 34, 114, 36, 5184, 38, 132, 40, 124026, 42, 150, 44, 160, 46, 934, 48, 1008, 50, 1084, 52, 43366048, 54, 1240

(list; graph; refs; listen; history; edit; text; internal format)

OFFSET

1,2

COMMENTS

A380891(x) map is If x mod 2 = 0 then a(x) = floor(x^(1/3)) else a(x) = floor(x^(4/3)).

LINKS

James C. McMahon, Table of n, a(n) for n = 1..10000

Vikram Prasad and M. A. Prasad, Estimates of the maximum excursion constant and stopping constant of juggler-like sequences, ResearchGate, 2025.

MATHEMATICA

fj[n_]:=If[Mod[n, 2]==0, Floor[Surd[n, 3]], Floor[n^(4/3)]]; a381246[n_]:=Max[Delete[FixedPointList[fj, n], -1]]; Array[a381246, 55]

PROG

(Python)

import sys

import gmpy2

sys.set_int_max_str_digits(0)

def floorJuggler(n):

a=n

max=n

while a > 1:

b=0

if a%2 == 0:

b1=gmpy2.iroot(a, 3)

b=b1[0]

else:

b1=gmpy2.iroot(a**4, 3)

b=b1[0]

a=b

if a > max:

max = a

return max

maxcount=0

for i in range (1, 100):

print (i, floorJuggler(i))

CROSSREFS

Cf. A380891, A383135.

Sequence in context: A117295 A235999 A093820 * A095400 A237264 A265560

Adjacent sequences: A381243 A381244 A381245 * A381247 A381248 A381249

KEYWORD

nonn,new

AUTHOR

James C. McMahon and Vikram Prasad, Apr 17 2025

STATUS

approved

A383135 Contribution to the OEIS (supporting a juggler sequence)

 

A383135

a(n) = number of iterations that n requires to reach 1 under the x -> A380891(x) map, or -1 if it never does.

0

0, 1, 2, 1, 3, 1, 5, 2, 3, 2, 3, 2, 4, 2, 4, 2, 6, 2, 4, 2, 6, 2, 13, 2, 13, 2, 8, 3, 8, 3, 10, 3, 10, 3, 3, 3, 10, 3, 5, 3, 10, 3, 5, 3, 5, 3, 6, 3, 5, 3, 5, 3, 17, 3, 5, 3, 5, 3, 17, 3, 3, 3, 3, 2, 12, 2, 3, 2, 5, 2, 5, 2, 12, 2, 3, 2, 7, 2, 3, 2, 7, 2, 7, 2

(list; graph; refs; listen; history; edit; text; internal format)

OFFSET

1,3

COMMENTS

A380891(x) map is If x mod 2 = 0 then a(x) = floor(x^(1/3)) else a(x) = floor(x^(4/3)).

LINKS

James C. McMahon, Table of n, a(n) for n = 1..10000

Vikram Prasad and M. A. Prasad, Estimates of the maximum excursion constant and stopping constant of juggler-like sequences, ResearchGate, 2025.

MATHEMATICA

fj[n_]:=If[Mod[n, 2]==0, Floor[Surd[n, 3]], Floor[n^(4/3)]]; a383135[n_]:= Length[ NestWhileList[fj, n, # != 1 &]] - 1; Array[ a383135, 84]

PROG

(Python)

import sys

import gmpy2

sys.set_int_max_str_digits(0)

def floorJuggler(n) :

a=n

count=0

while (a > 1) :

b=0

if (a%2 == 0) :

b1=gmpy2.iroot(a, 3)

b=b1[0]

count=count+1

else :

b1=gmpy2.iroot(a**4, 3)

b=b1[0]

count=count+1

a=b

return count

for i in range (1, 100) :

print (i, floorJuggler(i))

CROSSREFS

Cf. A380891, A381246.

Cf. A006577.

Sequence in context: A277120 A104725 A289079 * A249810 A257111 A011129

Adjacent sequences: A383132 A383133 A383134 * A383136 A383137 A383138

KEYWORD

nonn,new

AUTHOR

James C. McMahon and Vikram Prasad, Apr 17 2025

STATUS

approved

Math Without Numbers - a book review

In preparing and studying for this blog, I read as many math books as possible. I haven't posted book reviews in this blog before, but Math Without Numbers by Milo Beckman and illustrated by M Erazo deserves a very positive review. Beckman truly succeeds in walking the reader through many mathematical topics without using numbers. To do so, many real life examples are given to go with the concepts.

Several mathematical proofs are presented in a refreshing format- sometimes with easy-to-follow illustrations but never with numbers or formulas. 

One of my favorite sections gave a good introduction to topology. Another presents the incompleteness theorem of Gödel in a dialog format. A surprisingly clear analogy is offered on how the law of gravity is comparable to estimating the trade between two countries. As a fan of automata, I enjoyed an example demonstrating how the traffic phenomena of gapers' block can be modeled with simple rules. To cap off the tour through many subjects, Beckman delivers an understandable explanation of the Standard Model of particle physics.

milobeckman.com

Twitter milobela

See prior post for other math book and website recommendations.

Additional Thoughts on the Collatz Conjecture (or the 3n+1 Problem)

The Collatz Conjecture states that all sequences following its two simple rules will reach 1. The conjecture was introduced in a prior post, HOTPO (half or triple plus one). The two rules are:

Start the sequence with any positive integer,
1. If the number is even, divide by 2

2. If the number is odd, triple and add 1

Stop if the new number is one; if not, repeat the set of rules.

The sequences have been shown through numerical simulation to always reach 1 without a counterexample found. However, no one has proven that the sequence will always reach 1. Hence, this is a conjecture and not a proven theorem.

Additional Thoughts

While I don't have a contribution to add to a proof, I offer a probabilistic view of the sequence. 

First, define Known Terminator as a number, that once reached, will bring the sequence to 1. Common known terminators are powers of 2, multiples of 5 and any number reached in a sequence that reaches 1 as an investigator explores different starting numbers.

Next, for a given starting number, No,  or any number in a generated sequence, assume there is a 50%-50% probability of the number being even or odd. The next number, N1,  in the series can be expressed algebraically on a probabilistic, expected-value basis:

N1 = 0.50 (No/2) + 0.50 (3No + 1) = 1/4 No + 3/2No + 1/2 = 7/4 No + 1/2

An alternate formulation of rule 2 is:

If the number is odd, triple and add 1, then divide by 2 (so stated because because it is guaranteed that if N is odd, 3N + 1 is even).

With the alternate formulation, the new expected value of N1 becomes:

N1 = 0.50 (No/2) + 0.50 (1/2)(3No + 1) = 1/4 No + 3/4No + 1/4 = No + 1/4

Since both terms, 7/4 No + 1/2 and No + 1/4, are greater than No, we can now state, on an expected value basis, the sequence tends to increase until reaching a known terminator.

For anyone who has explored the conjecture, this is stating the obvious, but I wanted to share in case it inspires some additional progress. I'll be challenging my fellow Collatz fans, Andrew Marshall and Sandra Perez, to contribute their thoughts. 

In the news:

Recent Popular Mechanics article - Collatz

Update August 26, 2020 - Collatz article in Quanta Magazine 

Another article sharing the probalistic approach of my post: https://www.quantamagazine.org/why-mathematicians-still-cant-solve-the-collatz-conjecture-20200922/

HOTPO - Collatz Conjecture

(Image: Gemini created)

Instructions

Begin with any number (limited to positive integers: 1, 2, 3, 4…). Follow these two simple rules to obtain the next number in the sequence:
                Rule 1: If the number is even, divide it by 2
or
                Rule 2: If the number is odd, triple it and add 1

If this next number is 1, stop; that is the end of the sequence. For any number other than one, use Rules 1 and 2 above to obtain the next number in the sequence. Continue using the rules and build up the sequence until you end at the number 1. The name “HOTPO” is based on the initials of the rules “Half Or Triple Plus One.”

Example 1

Starting number: 4

Since 4 is even, divide by two. The next number in the sequence is 2.

Since 2 is even, divide by two. The next number in the sequence is 1.

We reached 1 so now the sequence ends. The full sequence was: 4, 2, 1.

Example 2

Starting number: 3

Since 3 is odd, triple it and add one. The next number in the sequence is 3 x 3 + 1 = 10.

Since 10 is even, divide by two. The next number in the sequence is 5.

Since 5 is odd, triple it and add one. The next number in the sequence is 3 x 5 + 1 = 16.

Since 16 is even, divide it by two. The next number is 8.

Since 8 is even, divide it by two. The next number is 4.

Since 4 is even, divide it by two. The next number is 2.

Since 4 is even, divide it by two. The sequence now ends with 1.
The full sequence was 3, 10, 5, 16, 8, 4, 2, 1.

The two examples above show how the HOTPO sequence can develop and grow. Sometimes the sequence ends very quickly and other times the sequence goes up and down but eventually settles to 1. The interesting thing is that no matter what number you start with, the sequence will end with 1.

Suggested Exercises

1.        Find the starting number under 10 that produces the longest sequence.

2.       Find the starting number under 10 that produces the shortest sequence.

3.       Find the starting numbers between 20 and 100 that produce the shortest and longest sequences.

4.       Compete with a friend, classmate or family member to find the starting number with the longest sequence.

Sequences made by other rules

The HOTPO rules always produce sequences that end in 1. This is not the case for sequences produced by other rules. Explore the sequences that are created by a different set of rules described below.

Start with any positive integer. To obtain the next number in the sequence:

Rule 1: If the number is even, divide by 2 (same rule as before)

Or

Rule 2: If the number is odd, triple it and add 3

If the new number is 1, stop; otherwise continue.

Examples:

Starting number: 2

The number is even, so divide by two. The next number is 1 so the sequence ends. The full sequence is simply 2, 1.

Starting number: 3

The number is odd, so triple and add 3. The next number is 3 x 3 + 3 = 12.

The number is even, so divide by 2. The next number is 6.

The number is even, so divide by 2. The next number is 3.

The number is odd, so triple and add 3. The next number is 3 x 3 + 3 = 12.

The number is even, so divide by 2. The next number is 6.

The number is even, so divide by 2. The next number is 3.

The sequence is now stuck in a cyclic loop. The full sequence is: 3, 12, 6, 3, 12, 6, 3, 12, 6, 3…

Exercises

5.       Try this “Half or Triple plus Three” sequence for other starting numbers up to 11. What results do you see?

6.       Explore other sequences by making your own rules.

7.       (For advanced students): Try the original HOTPO rules applied to negative numbers.

For parents, teachers and advanced thinkers - How does it work?

This lesson is based on the Collatz Conjecture which states that the HOTPO sequence will always end in 1 for all starting numbers that are positive integers. A conjecture is a mathematical statement that is generally accepted as true but has not been proven. A formally proven statement is called a theorem. While the Collatz Conjecture has not been proven, it has been confirmed to be true using computers for integers beyond 1,000,000,000. Monetary prizes are available for the person who can formally prove the Collatz statement to be true. If a single starting number was found for which the HOTPO sequence does not end with 1, then the Collatz statement would be disproven. However, given the very large numbers that have already been tested, finding such a starting number is unlikely.

For additional information:

An animation of the Collatz sequence is available at: http://gfredericks.com/sandbox/arith/collatz

Exercise key and comments:

1.        The length of the HOTPO sequence starting with 9 is 20 numbers in length:
9, 28, 14, 7, 22, 11, 24, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1

2.       The shortest sequence starts with 2 with just 2 numbers in length: 2, 1. (One might also consider the case of the starting number of 1 if you say you reached one before applying Rule 1 or 2; however, following the rules you would first triple one and add one to get a second number of 4 so the sequence should be: 1, 4, 2, 1 for a total of numbers in the sequence.)

3.       The shortest sequence for starting numbers 20 to 100 begins with 32: 32, 16, 8, 4, 2, 1.
The longest sequence for starting numbers 20 to 100 begins with 97 (it takes 118 steps to complete).

4.       Suggestions for competing with a friend or classmate:
a. find longest sequence with a time limit of 30 minutes (or other time of your choosing)
b. compete once without the use of calculators and once using calculators.

5.       The rule Half or Triple Plus Three produces the following sequences for the starting numbers of 1-11:
Starting Number:             Sequence:

1                                                                     1, 6, 3, 12, 6, 3, 12, 6, 3 …

2                                                                     2, 1

3                                                                     3, 12, 6, 3, 12, 6, 3…

4                                                                    4, 2, 1

5                                                                     5, 18, 9, 30, 15, 48, 24, 12, 6, 3, 12, 6, 3…

6                                                                    6, 3, 12, 6, 3…

7                                                                     7, 24, 12, 6, 3, 12, 6, 3…

8                                                                    8, 4, 2, 1

9                                                                    9, 30, 15, 48, 24, 12, 6, 3, 12, 6, 3…

10                                                                 10, 5, 18, 9, 30, 15, 48, 24, 12, 6, 3, 12, 6, 3…

11                                                                   11, 36, 18, 9, 30, 15, 48, 24, 12, 6, 3, 12, 6, 3…

12                                                                  12, 6, 3, 12…

13                                                                  13, 42, 21, 66, 33, 102, 51, 154 (and continues 23 more terms before cycling back to 33, 102, 51…)

6.       Suggestions for your own rules for creating sequences:
a. Keep to two rules based on the seed number (if even do this; if odd do that)
b. Explore the sequences created by your rules with larger and larger starting numbers until you see that the sequences end in cycles or the sequence of numbers gets larger without end.
c. Suggestions: Try Half or Triple Minus One (some starting numbers will terminate with one and some end in a cyclic loop).

7.       (Advanced) Applying the HOTPO rules to negative numbers produces cyclic results:

Starting Number:             Sequence:

-1                                            -1, -2, -1 …

-2                                            -2, -1, -2, -1 …

-3                                            -3, -8, -4, -2, -1, -2, -1 …

-4                                            -4, -2, -1, -2, -1 …

-5                                            -5, -14, -7, -20, -10, -5, -14, -7, 20, 10 …

Side note: Sandra Perez of Mesa College, San Diego, presented this paper at the MCRC in 2019: 

Update (6/8/2020):

As a challenge listed in Project Euler (see other post), I wrote a Python program to calculate long Collatz chains. For starting numbers under 10,000,000, the number 8,400,511 produces the longest chain at 685 entries which I list below beginning with the entry after the start number: [25201534, 12600767, 37802302, 18901151, 56703454, 28351727, 85055182, 42527591, 127582774, 63791387, 191374162, 95687081, 287061244, 143530622, 71765311, 215295934, 107647967, 322943902, 161471951, 484415854, 242207927, 726623782, 363311891, 1089935674, 544967837, 1634903512, 817451756, 408725878, 204362939, 613088818, 306544409, 919633228, 459816614, 229908307, 689724922, 344862461, 1034587384, 517293692, 258646846, 129323423, 387970270, 193985135, 581955406, 290977703, 872933110, 436466555, 1309399666, 654699833, 1964099500, 982049750, 491024875, 1473074626, 736537313, 2209611940, 1104805970, 552402985, 1657208956, 828604478, 414302239, 1242906718, 621453359, 1864360078, 932180039, 2796540118, 1398270059, 4194810178, 2097405089, 6292215268, 3146107634, 1573053817, 4719161452, 2359580726, 1179790363, 3539371090, 1769685545, 5309056636, 2654528318, 1327264159, 3981792478, 1990896239, 5972688718, 2986344359, 8959033078, 4479516539, 13438549618, 6719274809, 20157824428, 10078912214, 5039456107, 15118368322, 7559184161, 22677552484, 11338776242, 5669388121, 17008164364, 8504082182, 4252041091, 12756123274, 6378061637, 19134184912, 9567092456, 4783546228, 2391773114, 1195886557, 3587659672, 1793829836, 896914918, 448457459, 1345372378, 672686189, 2018058568, 1009029284, 504514642, 252257321, 756771964, 378385982, 189192991, 567578974, 283789487, 851368462, 425684231, 1277052694, 638526347, 1915579042, 957789521, 2873368564, 1436684282, 718342141, 2155026424, 1077513212, 538756606, 269378303, 808134910, 404067455, 1212202366, 606101183, 1818303550, 909151775, 2727455326, 1363727663, 4091182990, 2045591495, 6136774486, 3068387243, 9205161730, 4602580865, 13807742596, 6903871298, 3451935649, 10355806948, 5177903474, 2588951737, 7766855212, 3883427606, 1941713803, 5825141410, 2912570705, 8737712116, 4368856058, 2184428029, 6553284088, 3276642044, 1638321022, 819160511, 2457481534, 1228740767, 3686222302, 1843111151, 5529333454, 2764666727, 8294000182, 4147000091, 12441000274, 6220500137, 18661500412, 9330750206, 4665375103, 13996125310, 6998062655, 20994187966, 10497093983, 31491281950, 15745640975, 47236922926, 23618461463, 70855384390, 35427692195, 106283076586, 53141538293, 159424614880, 79712307440, 39856153720, 19928076860, 9964038430, 4982019215, 14946057646, 7473028823, 22419086470, 11209543235, 33628629706, 16814314853, 50442944560, 25221472280, 12610736140, 6305368070, 3152684035, 9458052106, 4729026053, 14187078160, 7093539080, 3546769540, 1773384770, 886692385, 2660077156, 1330038578, 665019289, 1995057868, 997528934, 498764467, 1496293402, 748146701, 2244440104, 1122220052, 561110026, 280555013, 841665040, 420832520, 210416260, 105208130, 52604065, 157812196, 78906098, 39453049, 118359148, 59179574, 29589787, 88769362, 44384681, 133154044, 66577022, 33288511, 99865534, 49932767, 149798302, 74899151, 224697454, 112348727, 337046182, 168523091, 505569274, 252784637, 758353912, 379176956, 189588478, 94794239, 284382718, 142191359, 426574078, 213287039, 639861118, 319930559, 959791678, 479895839, 1439687518, 719843759, 2159531278, 1079765639, 3239296918, 1619648459, 4858945378, 2429472689, 7288418068, 3644209034, 1822104517, 5466313552, 2733156776, 1366578388, 683289194, 341644597, 1024933792, 512466896, 256233448, 128116724, 64058362, 32029181, 96087544, 48043772, 24021886, 12010943, 36032830, 18016415, 54049246, 27024623, 81073870, 40536935, 121610806, 60805403, 182416210, 91208105, 273624316, 136812158, 68406079, 205218238, 102609119, 307827358, 153913679, 461741038, 230870519, 692611558, 346305779, 1038917338, 519458669, 1558376008, 779188004, 389594002, 194797001, 584391004, 292195502, 146097751, 438293254, 219146627, 657439882, 328719941, 986159824, 493079912, 246539956, 123269978, 61634989, 184904968, 92452484, 46226242, 23113121, 69339364, 34669682, 17334841, 52004524, 26002262, 13001131, 39003394, 19501697, 58505092, 29252546, 14626273, 43878820, 21939410, 10969705, 32909116, 16454558, 8227279, 24681838, 12340919, 37022758, 18511379, 55534138, 27767069, 83301208, 41650604, 20825302, 10412651, 31237954, 15618977, 46856932, 23428466, 11714233, 35142700, 17571350, 8785675, 26357026, 13178513, 39535540, 19767770, 9883885, 29651656, 14825828, 7412914, 3706457, 11119372, 5559686, 2779843, 8339530, 4169765, 12509296, 6254648, 3127324, 1563662, 781831, 2345494, 1172747, 3518242, 1759121, 5277364, 2638682, 1319341, 3958024, 1979012, 989506, 494753, 1484260, 742130, 371065, 1113196, 556598, 278299, 834898, 417449, 1252348, 626174, 313087, 939262, 469631, 1408894, 704447, 2113342, 1056671, 3170014, 1585007, 4755022, 2377511, 7132534, 3566267, 10698802, 5349401, 16048204, 8024102, 4012051, 12036154, 6018077, 18054232, 9027116, 4513558, 2256779, 6770338, 3385169, 10155508, 5077754, 2538877, 7616632, 3808316, 1904158, 952079, 2856238, 1428119, 4284358, 2142179, 6426538, 3213269, 9639808, 4819904, 2409952, 1204976, 602488, 301244, 150622, 75311, 225934, 112967, 338902, 169451, 508354, 254177, 762532, 381266, 190633, 571900, 285950, 142975, 428926, 214463, 643390, 321695, 965086, 482543, 1447630, 723815, 2171446, 1085723, 3257170, 1628585, 4885756, 2442878, 1221439, 3664318, 1832159, 5496478, 2748239, 8244718, 4122359, 12367078, 6183539, 18550618, 9275309, 27825928, 13912964, 6956482, 3478241, 10434724, 5217362, 2608681, 7826044, 3913022, 1956511, 5869534, 2934767, 8804302, 4402151, 13206454, 6603227, 19809682, 9904841, 29714524, 14857262, 7428631, 22285894, 11142947, 33428842, 16714421, 50143264, 25071632, 12535816, 6267908, 3133954, 1566977, 4700932, 2350466, 1175233, 3525700, 1762850, 881425, 2644276, 1322138, 661069, 1983208, 991604, 495802, 247901, 743704, 371852, 185926, 92963, 278890, 139445, 418336, 209168, 104584, 52292, 26146, 13073, 39220, 19610, 9805, 29416, 14708, 7354, 3677, 11032, 5516, 2758, 1379, 4138, 2069, 6208, 3104, 1552, 776, 388, 194, 97, 292, 146, 73, 220, 110, 55, 166, 83, 250, 125, 376, 188, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668, 334, 167, 502, 251, 754, 377, 1132, 566, 283, 850, 425, 1276, 638, 319, 958, 479, 1438, 719, 2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644, 1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232, 4616, 2308, 1154, 577, 1732, 866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1]