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 Le Tour Des Bases, Visit each number base; try them out
icarus
  Posted: May 1 2012, 03:01 PM


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Le Tour des Bases

Get your road gear on, we’re pedaling into the hills visiting various number bases along the way! This forum and the Dozenal Societies often discuss number bases, especially base twelve. Here, we’ll cruise through the main features of each number base. Each tour consists of a brief “travelogue” from the standpoint of general human computation — what would it be like to actually use the number base on a daily basis? The travelogue refers you to “Icarus’s Standard Nomenclature for Number Bases”, a sort of “road map” explaining some of the sights along the way. Like a phrasebook for foreign travel, or a guidebook describing the kind of attractions you’ll likely see in every city on your trip, the map is important because it defines some of the terms used in the narrative. We need to know a few things common to all the number bases so that we can describe their properties “apples to apples”. We’ll dive into the digits of the number base, as all number bases represent the world through their digits. A short synopsis of the kinds of digits and their relationships to the base follows; digits are like the main attractions in a town. Here, some key terms link to expanded descriptions and references to maths texts so you can make sure we’re not taking you for a ride. Next, we’ll look at the addition and multiplication tables, basic tools of computation. Here you can “test drive” maths in the number base — try your hand at memorizing the facts. Running some arithmetic given these tables, you can get a little sense of what it’s like to “live” in the base we’re visiting. Fractions are a key consideration when thinking about a good number base. Will the base help us with fractions, or will it just make things more difficult? We’ll look at fractions in four different ways. First, we take a look at reciprocals of the smallest integers (example, the reciprocal of 2 is ½). Everyone loves fractions that have short terminating digital representations, e.g., ½ = 0.5 decimal. The second exploration of fractions examines the primes less than 30 to see whether their reciprocals have terminating or recurrent digital representations. If the prime reciprocal has a recurrent digital representation, the table lists the length compared to the maximum length, and the reptend (group of numbers repeated), e.g., one seventh decimally has a reptend length of 6 out of a maximum length of 6; it repeats “142857”. The third table looks at reciprocals of coprimes with the shortest 8 reptends. (Coprimes are integers “out of phase” with the base, such as 3 or 7 in base 10). The final fraction-related table looks at the reciprocals of regular numbers smaller than the cube of the base. (Regular numbers are integers whose reciprocals terminate when expressend in the base in question, such as 4 or 25 in base 10). Our tour takes us through intuitive divisibility rules for the number base, then a quick shot through logarithms for each digit and the values of some irrational numbers.

Oftentimes you may read claims on this forum and in the writings at the Dozenal Societies of America and Great Britain. Dozenalists tend to examine number bases more closely than many accomplished mathematicians, as number bases are an elementary form of mathematics. Their writings tend to incorporate notions with which the writers are familiar with in several bases, but with which many readers are not at all familiar. This tour is devised to let you examine the main properties of each number base so you can form your own opinion. The travelogues are indeed written from the perspective of someone looking for a base to facilitate computation and are a sort of introduction, but the data that follow it derive from elementary number theory, which anyone can test. It is my endeavour to maintain a fair tour, especially in the data portion of the visit, but the introductions indeed work like a travel guide celebrating the noble features of a base as well as some potential pitfalls you might encounter during your trip.

I hope you enjoy the Tour des Bases! If you get a “flat tire” or spot a flaw in the route, post something and we’ll be happy to chat about it or amend the route. If you get winded on the hills, take a break and join us another day. Responses are welcome, as well as additional information you’d like to post, after all, this is a forum! Get your helmets, fill those water bottles, and pack a spare tube, let’s roll! (Start the tour with base 6, 8, 10, 11, 12, 14, 15, 16, or 20; new tours will be posted now and then).

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icarus
Posted: May 1 2012, 03:04 PM


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Le Tour des Bases · List of Tours

Links to tours of number bases follow. Note that tours to new bases will be posted from time to time. I’ve written notes for bases between 2 and 30, all of which will eventually find their way on the tour.

Base 2 • Busy Binary · Keep It Simple, Smily!
Base 3 • Trusty Little Ternary · Forward, neutral, and reverse.
Base 4 • Cute Questing Quaternary · May the fours be with u
Base 5Quick Kid: Quinary · Give him a helping hand!
Base 6Hey Superior Senary, You Ready to Rumble? · In this corner, the featherweight six!

“Human Scale” Bases
Base 7Lucky, Sacred Seven ·Prime right in the sweet spot
Base 8Octal, the Awkward Beauty · Sure she’s pretty but man what a klutz!
Base 9Everything Done up to the Nines · What a square!
Base 10Plain Ol’ Vanilla Decimal · Very handy, he’s the king of the world!
Base 11Undecimal the Unbelievable · Not decimal indeed!
Base 12Delightful Darling Dozenal · Sublime simplicity (with a couple flaws)
Base 13Tridecimal Phobia · Be afraid, be very afraid…
Base 14Operation Base Fortnight · Let’s explore base fourteen!
Base 15La Quinceañera · Try out base fifteen!
Base 16Sympathy for the Hexadecimal · Everyone has their first love…

Lower Mid-Scale Bases
Base 17Seventeen the Dancing Queen · Settle down, she's all air upstairs
Base 18Octodecimal, Not Quite Semiprime, but… · If you like 9, you’ll love her sister
Base 19Product Enneadecimal · Annual sale, cash only, no returns
Base 20Vigesimal, Super-Decimal on Steroids? · Take off yer shoes, we’re counting to 20!
Base 21Unvigesimal Light and Magic · Three's a Charm Meets Lucky Seven
Base 24Acrobatic Tetravigesimal (Base 24) · Sticking her landings (when needed)
Base 25Bingo Pentavigesimal (Base 25) · Free chip on the fives, buster.
Base 2828: Too much water weakens the tea! · A little bit weaker now…
Base 30Outpost Trigesimal · Balanced tricolor detector at the edge
Base 32Duotrigesimal · Leaving the domain of two · Courtesy of Double Sharp
Base 34Agent 34: Prime Detective · Our most nimble prime agent
Base 3535: Minimally Distinct but Obscure · Where 6, 15, 34, and 36 relate
Base 3636: A Marriage of Squares! · You think her parents were squares…
Base 4040: Quadragesimal · Courtesy of Double Sharp
Base 4242: The Answer to the Universe (Not) · If you prefer 7 vs. 5, try me!
Base 4848: You got your 3 in my hexadecimal! · Say, this 3-smooth base tastes great!
Base 54Tetraquinquagesimal · 3-smooth strength sapped by totatives · Courtesy of Double Sharp
Base 55Agent 55: Divisibility Detective · Understudy to Agent 34
Base 5656: Hexaquinquagesimal · Courtesy of Double Sharp

Upper Mid-Scale Bases
Base 60Sublime Sexagesimal · Tailor-Made for Aliquot Division
Base 64Base 64 · The Checkerboard Base · Courtesy of Double Sharp
Base 70Sweet Sister 70 · Unexpected Rival to the Near Misses
Base 72Silly 72 Sitting on a Post · What Happened, Girl?
Base 80Base 80, Hexadecimal <3s Quinary · and Gets Along with His Brother 3
Base 8484, Alternative to Sexagesimal? · Johnny-Come-Lately?
Base 90Base 90: Plenty Fine · A runner-up that does quite well
Base 96Base 96: Midscale 3-smooth Social Butterfly · The middle sister with all the friends
Base 99Agent 99 · A Thorough Prime Detective
Base 100The Hundred as a Number Base · Everybody loves 100%!
Base 108108: Three cubed, two squared · Hey man, that 3-smooth tune is so lame
Base 112112: Rarified Air at the Top of Mid-Scale · Captain take 'er down a notch
Base 120120: The Great Hundred · King of Direct and Indirect Properties
The Dozen Mashup · Comparison of Bases 12k with 1 ≤ k ≤ 12 (Figures are dozenal in the thread)
The Primorial Mashup · Comparison of Primorial Bases
Comparison of Bases 60 and 120 · Godzilla vs. Mothra!
Midscale Mashup · Comparison of bases between 60 and 120

Grand Bases
Base 109109: At least the neighbors are nice · Large prime example with keen neighbors
Base 144144: Twelve’s Bigger Brother · Come for direct, stay for indirect properties
Base 126Perils of a too-good neighborhood · Courtesy of Double Sharp
Base 168The tetradecimal &dquo;short hundred” · Courtesy of Double Sharp
Base 173173: Cold Shoulders from the Neighbors · Large prime example with bad neighbors
Base 180Base 180: Time to Turn Around? · Courtesy of Double Sharp
Base 210Base 210: Diminishing Returns? · Reducing totient ratio at expense
Base 240240: Pounds, Shillings, & Pence · The last bastion of real utility?
Base 351351: Grand Dame of Extrinsic Strength · Solid walls of divisibility tests!
Base 360Superhero 360 · Aid to angles but sapped by kryptonite
Base 4202² · 3 · 5 · 7 · An alternative to 360?
Base 480A Traditional Ream Of Paper, though it’ll take more for the tables · Courtesy of Double Sharp
Base 504Quincentoquadral · The tetradecimal circle of degrees · Courtesy of Double Sharp
Base 720720, Goliath · A dozenal unit circle auxiliary
Base 8402³ · 3 · 5 · 7

“Randomly Considered” Bases
Base 27 • Heptavigesimal.
Base 29 • Enneavigesimal.
Base 40 • Quadragesimal.
Base 49 • Enneaquadragesimal. (Thanks oschkar!)
Base 51 • Unquinquagesimal.
Base 75 • Pentaseptuagesimal.
Base 82 • Duoöctagesimal.
Base 85 • Pentoctagesimal.
Base 119 • Centenneadecimal.
Base 126 • Centohexavigesimal.
Base 133 • Centotritrigesimal. (Base “Lee”)
Base 169 • Centenneasexagesimal.
Base 289 • Ducentenneöctagesimal.
Base 504 • Quincentoquadral.
Base 960 • Novocentosexagesimal.

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icarus
Posted: May 1 2012, 03:14 PM


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Map of the Tour

This map arranges each number base between 2 and 30 inclusive in rows, the base listed in the leftmost column. Each digit of each base appears on the same row as the listed base. The decimal values of the base and the digits are given for your convenience. Read more about each type of digit here.

      Digit Maps for Bases 2 ≤ r ≤ 30
2 0 1    
3 0 1 2    
4 0 1 2 3    
5 0 1 2 3 4    
6 0 1 2 3 4 5    
7 0 1 2 3 4 5 6    
8 0 1 2 3 4 5 6 7    
9 0 1 2 3 4 5 6 7 8    
10 0 1 2 3 4 5 6 7 8 9    
11 0 1 2 3 4 5 6 7 8 9 10    
12 0 1 2 3 4 5 6 7 8 9 10 11    
13 0 1 2 3 4 5 6 7 8 9 10 11 12    
14 0 1 2 3 4 5 6 7 8 9 10 11 12 13    
15 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14    
16 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15    
17 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16    
18 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17    
19 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18    
20 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19    
21 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20    
22 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21    
23 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22    
24 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23    
25 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24    
26 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25    
27 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26    
28 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27    
29 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28    
30 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29  
  0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Legend

Unit, 1 | r ∧ 1 ⊥ r, more

Divisor, d | r, more

Semidivisor, g = ∏ piδ ∧ (δ > ρ) ∧ (g ¬| r) ∧ (0 < gr), more

Semitotative, h = ∏ piqj ∧ (0 < h < r), more

Totative (opaque), tr ∧ 0 < t < r, more

α-Totative, tαr ∧ , tα | (r + 1), more

ω-Totative, tωr ∧ , tω | (r − 1), more

α/ω-Totative, tαtω, more

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Dan
Posted: May 2 2012, 12:39 AM


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QUOTE (icarus @ May 1 2012, 09:04 AM)

Le Tour des Bases · List of Tours

Links to tours of number bases follow. Note that tours to new bases will be posted from time to time. I’ve written notes for bases between 2 and 30, all of which will eventually find their way on the tour.

Base 6Hey Superior Senary, You Ready to Rumble? · In this corner, the featherweight six!
Base 8Octal, the Awkward Beauty · Sure she’s pretty but man what a klutz!
Base 9 • Everything Done up to the Nines · What a square!
Base 10Plain Ol’ Vanilla Decimal · Very handy, he’s the king of the world!
Base 11Undecimal the Unbelievable · Not decimal indeed!
Base 12Delightful Darling Dozenal · Sublime simplicity (with a couple flaws)
Base 13 • Tridecimal Phobia · Be afraid, be very afraid…
Base 14Operation Base Fortnight · Let’s explore base fourteen!
Base 15La Quinceañera · Try out base fifteen!
Base 16Sympathy for the Hexadecimal · Everyone has their first love…
Base 18 • Octodecimal, Not Quite Semiprime, but… · If you like 9, you’ll love her sister
Base 20Vigesimal, Super-Decimal on Steroids? · Take off yer shoes, we’re counting to 20!


I'm curious as to how you decided that 10 is male but 8, 9, and 18 are female.
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icarus
Posted: May 2 2012, 02:06 AM


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Dan,

Indeed there is a pattern! Even if it is accidental… (recalling from memory…) I speak Russian, that should be a hint, and the pattern is number theoretical at least with part of the set of possible categories.
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Dan
Posted: May 2 2012, 03:45 AM


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I don't know Russian, so the hint is lost on me.
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dgoodmaniii
Posted: May 2 2012, 06:08 PM


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QUOTE (Dan @ May 2 2012, 03:45 AM)
I don't know Russian, so the hint is lost on me.

Grammatical gender. I don't speak (much) Russian, so I can't so for sure, but I think he's ascribing sex to these numbers based on their grammatical gender in Russian.
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icarus
Posted: May 2 2012, 09:11 PM


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Gentlemen,

Actually all I meant was I think the assignment of gender is completely random but there are three of them; feminine {(6), 8, 9, 12, 15, (16), 18} masculine {(6), 10, (20)}, and neuter {7, 11, 13, 14, 17}. So it appears that powers of two, multiples of three are feminine, multiples of ten are masculine, and primes and multiples of 7 are neuter.

I say 6 can be masculine or feminine, maybe stretching it, because my wife is into MMA. So women can be fighters, tho I am not sure the whole "welterweight" deal applies to them.

Anyway, the entire masculine/feminine/neuter thing is entirely random, just there to make the stories a little more lively. The 8 "because it has more beauty" had me thinking female, the "beauty" remark comes from the poster of the thread and not me. Sixteen may not be female from the title, as the title refers to a Stones song I happened to be listening to at the time: "Please allow me to introduce myself / I'm a man of wealth and taste...". No, the masculine/feminine/neuter gender trends won't necessarily be carried up or down scale; 21 may be male or neuter or something else. I think I am trying to impose order given dan's question. No this isn't an identity politics statement either. It's simply for levity.

I do hope you enjoy the posts. This is simply a proof of concept for a particular audience that isn't necessarily on the forum, or at least active on it. It also happens to serve to illustrate the properties of other number bases for folks who'd like to see things for themselves (skeptics/sceptics).
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icarus
Posted: Aug 29 2012, 02:15 PM


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Folks,

I've been able to take my multiplication tables from a print document and fold them into HTML so now I can produce tours of mid-scale bases pretty easily. I am working on a method of transferring Mathematica tables to Excel. When this happens I can produce any size table, though the notation won't be the same, instead it would be decimal coded. I think we can continue to explore the very high bases if desired.

Others have written great pieces on other bases. This series will continue to expand in the same character as the previous essays, with links to the other conversations (i.e., tridecimal, base-24, base-22, etc.). This way this forum will eventually have the broadest such number base tour on the internet, if it isn't already so.

I may revisit a few earlier threads and inject some new data so that they are on par with the latest essays. If there is anything (practical!) you'd like to add in the tour of each base, please suggest it.

Again, the frequent posters on this site continue to influence me in seeing the "sights" to be seen in these number bases. Please don't wait till I cover a given base to chat about it. I can mention a "sight" and link to whatever conversation has been had about it in other threads, so that we have an integrated experience.

I've written most of bases 9, 24, and 21; these will come next. I may try to write 60 or 120, but want to do my "experiment" first. As we push into September, I may not have much time to write, so the progress will be discontinuous.
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icarus
Posted: Aug 31 2012, 01:58 PM


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I've updated decimal, dozenal, and hexadecimal threads to show "regular figures", which are equivalent to "units" defined by Wendy Krieger at her "Number Theory 102" thread in the number theory forum. Also, I've added diagrams of patterns in the decimal, dozenal, and hexadecimal multiplication tables, and diagrams showing the relationship of the positive primes less than the base in the same threads.

There is a limit to the size of posts and this has affected the hexadecimal thread the most. The usual format is interrupted, but the same data is linked from the introductory post.
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icarus
Posted: Sep 6 2012, 09:27 PM


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I've added some limited tours of "grand bases" 210, 240, and 360. These were laying around this summer in partial form when I was mulling over other matters and away from the forum. If you've got a "grand base" to inspect, I'll put it on the queue. I am producing base 2520 but that's experimental and we'll see what needs to be done to do it. Hopefully some of these might spark some debate. I think they are curiosities, mountains that are fun to climb but impractical places to live.

Look at the tour menu, second post in this thread.
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Dan
Posted: Sep 7 2012, 12:34 AM


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While it may be a little too small to count as a "grand base", I'd like to see an analysis of Base 36, the highest base supported by the C strtol function (due to the convention of using 0-9 and A-Z for digits), and coincidentally a highly-composite number.
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Oschkar
Posted: Sep 7 2012, 01:08 AM


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QUOTE (Dan @ Sep 7 2012, 12:34 AM)
While it may be a little too small to count as a "grand base", I'd like to see an analysis of Base 36, the highest base supported by the C strtol function (due to the convention of using 0-9 and A-Z for digits), and coincidentally a highly-composite number.

I would like to see 36, 40, 42, 48, 56, 60, 72 and 84 as possible midscale bases on the way to grand 120 and higher.
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icarus
Posted: Sep 7 2012, 02:41 AM


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I wrote about base 144 tonight, and it ties to many posts on this forum (Shaun, m1n1f1g et al.) in a cool way. Will do 100 too. These things tend to illustrate a tidbit, but aren't terribly useful. Then 36 will be next.

I have tables for 36 and 60. I am not sure the board will handle the tables, but maybe I could split them. I've dispensed with the tables for "grand bases", but may circle back and do other things done for the small bases. A link to a PDF for bases larger than 40 or so might relieve the board of so many big tables. This stuff can fold over into indesign to be a big book of bases one day, that's exciting (to me).

Indeed many of the bases Oshkar suggested are on the list. I had HCNs, and the "runner up" bases to the HCNs, I think they are {24, 30, 36, 48, 60, 72, 84, 90, 96, 120, 144, 168, 180} plus Wendy's/Triesaran's beneficial-flank numbers {21, 34, 55, 99, 120} (doing from memory). Will add 42, 56. I'd written the full list on the menu post but commented them out. (it looked daunting! And I didn't want to promise and leave it stand) I also felt averse to starting a bunch of threads, only to have a "partial" set. Now I think we start threads and folks can comment and see patterns etc., with other info coming later. I think I'll start the bases off like the grand bases then fill in the details later. If you've written something about the bases please post a link to your discussions in the tour thread so people reach your thoughts, if I hadn't done that already. I am not as familiar with what happened between May and mid August this year, due to all the mulling on another non-number-base issue. (my midlife crisis, lol. Good thing: it's mostly over!)

Any idea on a really big prime (between say 60 and 360), just to illustrate what many of these look like? I think 55 will serve as a really big semi prime.
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Dan
Posted: Sep 7 2012, 02:50 AM


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QUOTE (icarus @ Sep 6 2012, 09:41 PM)
Any idea on a really big prime (between say 60 and 360), just to illustrate what many of these look like?

61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359

Pick one.
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wendy.krieger
Posted: Sep 7 2012, 07:50 AM


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70 has a place in there somewhere. This is the first abundant number that can not be represented as the sum of its divisors. Also, i am rather fond of it. It has, for example, among the squares, 2.00.01.

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Oschkar
Posted: Sep 7 2012, 08:37 PM


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QUOTE (Dan @ Sep 7 2012, 02:50 AM)
QUOTE (icarus @ Sep 6 2012, 09:41 PM)
Any idea on a really big prime (between say 60 and 360), just to illustrate what many of these look like?

61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359

Pick one.

In this case, I'd choose 109, probably, for its intutive relationships with 108=2^2x3^3 and 110=2x5x11. It is not exactly the typical large prime base, but it is rather versatile for such a base. On the other hand, 173 is an example of the worst possible type of prime base, with 172=2^2x43 and 174=2x3x29 as neighbors. You could do both as contrasting bases.

And I suggest that you separate 60 from 120.
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icarus
Posted: Sep 7 2012, 10:04 PM


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Oschkar: surely we will separate bases 60 and 120: that was a comparison and a stand-in till I get a proper page set up.

I like the suggestion 109 and 173 for the very reasons you've described.

The weird thing that occurred to me is with any composite, I can "shorten" the table by using two divisors. For the primes, the "shortening" will look odd, because the numbers don't have nontrivial integer divisors. I am trying to respect 30 cells as the maximum width, so might use some convenient number below that to "fold" the cells.

Wendy: 70 is now on the list!
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m1n1f1g
Posted: Sep 8 2012, 11:16 PM


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QUOTE (icarus @ Sep 7 2012, 03:41 AM)
I have tables for 36 and 60.

What about complementary divisor method ("truncated") tables? I'd like to see what they're like in terms of size and memorability.
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Treisaran
Posted: Sep 9 2012, 12:25 AM


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Very impressive, Icarus! (Yeah, mood's improved a bit the past few days smile.gif ) I'm especially interested in the rundowns of those bases that have a relationship to the dozen: 6 (half-dozen; already here), 18 (dozen half-dozen, or as I call it, dozen ha'zen wink.gif ), 24 (double dozen), 36 (triple dozen), 96 (octodoz?) etc. The last three I find intriguing because of the slight improvements (at an undeniable price, of course) they offer over dozenal, through their neighbour relations (96 is for the 240-lovers who want something better than a primeflank but don't want to abandon the sixteenfold division). All in contrast to 18, that 'total loss' of near-dozenal bases, which gains nothing and loses quite a lot.

On another note, I think a comparison of 34 and 120 is warranted: together with their neighbour relationships, they offer the same factors:
  • 34 = 2·17, ω = 3·11, α = 5·7
  • 120 = 2·2·2·3·5, ω = 7·17, α = 11·11

and yet, they don't have the same utility. The diprime 34 is at a considerable disadvantage in comparison to the divisor-rich 120. That's another case for the argument that handy neighbour relationships are no substitute for richness in divisors.
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icarus
Posted: Sep 9 2012, 02:05 AM


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M1n1f1g, I'll add abbreviated tables soon, till then here are AMTs for bases 10-56 and 60-120 at my web portfolio. treisaran thanks for the feedback! Indeed I haven't posted base 18 yet, and have skipped the mid scale bases thus far. I have started base 2520, a leviathan, will take a couple weeks, jobs rolling in soon. Indeed I am trying to integrate the tour with your thoughts as well as those of others. I'll also direct folks to the SPD test in the 144 and 12 entries.
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Oschkar
Posted: Sep 9 2012, 03:49 AM


Dozens Disciple


Group: Members
Posts: 575
Member No.: 623
Joined: 19-November 11



To me, the relationship between 21 and 22 looks a lot like the one between 14 and 15, only backwards, since 10, 14 and 22 are all manageable semiprimes. For example, like 10 uses the omega for 3, 14 the alpha, and 22 the omega again. Bases 10 and 14 both are compatible with 5, and 22 could use SPD to reduce the numbers with the alpha-2, since duovigesimal 101=decimal 445 is divisible, and there are still only 88 multiples to memorize (easily reduced to 22 by subtracting multiples of duovigesimal 50). For 7, 14 has it as a divisor, and in 22 it is an omega inheritor, but in decimal, 7 is out of SPD reach (143 multiples of 7 less than the decimal thousand, and subtracting multiples of 70, or even of 98, is rather impractical for a 3 digit number).
Although now that I think about it, the fact that 301 is so close to 300 could be used as an extension of SPD in decimal (not a neighbor of the square of the base, but of a multiple of it). To test for 7, instead of adjusting the remaining digits by what was added to the final 2, adjust them by triple that amount:
164472424704
1644724247 04+3=7 → 47+9=56
16447242 56
164472 42
1644 72-2=70 → 44-6=38
16 38-3=35 → 16-9=7
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Treisaran
Posted: Sep 9 2012, 01:50 PM


Dozens Disciple


Group: Members
Posts: 1,221
Member No.: 630
Joined: 14-February 12



QUOTE (icarus)
I have started base 2520, a leviathan, will take a couple weeks, jobs rolling in soon.


Good luck! But how are you going to squeeze the huge tables required for this monstrosity of a base into forum posts? dry.gif

QUOTE (Oschkar)
and there are still only 88 multiples to memorize (easily reduced to 22 by subtracting multiples of duovigesimal 50).


Compacting the number of SPD sequences, eh? Sounds good. I just wonder how far it could be carried. For base 22 it looks feasible, but I wonder about base 32, or the dozenal SPD test for 7 (which is based on *1001, the dozenal cube-α).

QUOTE
and 22 could use SDN [...] 7 is out of SDN reach [...] could be used as an extension of SDN in decimal


SDN stands for 'Systematic Dozenal Nomenclature', a comprehensive dozenal number naming scheme devised by Kodegadulo on this board; SPD stands for 'Split, Promote, Discard', a divisibility testing shortcut method stumbled upon by yours truly. smile.gif

Reminds me of that passage from the 1989 film Hunt for Red October: 'Pavarotti is a singer, Paganini was a composer'. smile.gif
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Oschkar
Posted: Sep 9 2012, 05:32 PM


Dozens Disciple


Group: Members
Posts: 575
Member No.: 623
Joined: 19-November 11



QUOTE (Treisaran @ Sep 9 2012, 01:50 PM)
QUOTE
and 22 could use SDN [...] 7 is out of SDN reach [...] could be used as an extension of SDN in decimal


SDN stands for 'Systematic Dozenal Nomenclature', a comprehensive dozenal number naming scheme devised by Kodegadulo on this board; SPD stands for 'Split, Promote, Discard', a divisibility testing shortcut method stumbled upon by yours truly. smile.gif

Reminds me of that passage from the 1989 film Hunt for Red October: 'Pavarotti is a singer, Paganini was a composer'. smile.gif

Thanks. Fixed.
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