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Pages: (10) « First ... 6 7 [8] 9 10  ( Go to first unread post )

 Le Tour Des Bases, Visit each number base; try them out
icarus
Posted: Dec 5 2016, 02:59 PM


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My opinion is that SDN is what it is "objectively"; how it is locally pronounced is not something we can control. The Lamadrid system by contrast was intended to build a scalable "vernacular" base name system. Thus I think SDN remains as is but we should attempt, in any language we support, to build base names that seem already part of that language.
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icarus
Posted: Dec 5 2016, 03:59 PM


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Wendy:

Added it! (see the register update log post).
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Kodegadulo
Posted: Dec 5 2016, 04:20 PM


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In modern Greek (I'll transliterate): kilometer is khiliometro, literally "thousandometer"; centimeter is ekatostometro, literally "hundredthometer"; millimeter is khiliostometro, literally "thousandthometer". So there is some precedent for taking a lot of liberty with a prefix system originally designed for another language (in this case, French). (Do we really believe the French Metricists were thinking of anybody other than gallophones? I bet they were expecting Academy French to be everybody's lingua franca ad infinitum.)

SDN really only tried to cater to English, but we kind of expect other languages will take some liberties with its pronunciation.
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Double sharp
Posted: Dec 24 2016, 01:29 PM


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I had a bit of time, so I did another mid-scale "guest post": here's base 54.

There's not really that many other bases that I think are particularly interesting to cover above 60. The exceptions are {168} (as the tetradecimal "twelfty") and {180} (as another step in Robin's inequality), but they are really too big. Perhaps {336} would be nice as well to complete the sequence of largely composite numbers up to 360.

Below 60, we might benefit from the addition of the bases {22, 26, 27, 32, 45, 50}, as well as of course the fundamental {2, 3, 4}.
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Oschkar
Posted: Dec 27 2016, 11:22 PM


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Another Mathematica request, icarus:

Can you calculate e^gamma*n*ln(ln(n))/sigma(n), where gamma is the Euler-Mascheroni constant and sigma(n) is the sum-of-divisors function, for all n from 2 to 60480, and post the values of n such that that function gives a result less than 5/4?
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Oschkar
Posted: Dec 29 2016, 01:06 AM


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Never mind; I did it in Python in a few minutes. The earliest bases in my list not yet covered by the Tour are 180 (already suggested by Doublesharp), and the three-digit alternating bases 2520, 5040 and 1680. Discarding bases larger than 840 thereafter, we have 168 (the tetradecimal long-hundred analogue), 480 (a traditional ream of paper), 504, 540 (one and a half turns), 252, 336 (the tetradecimal 240 analogue), 600 (two metric feet), 300 (a metric foot) and 216 (the senary thousand). Out of these, only {216, 252, 300} arenít largely composite, though they are incredibly convenient numbers that would have widespread uses as auxiliary bases.

CODE
(0, -inf),
(1, -inf),
(2, -0.4351907024566895),
(3, 0.1256294938049997),
(4, 0.3324338692674033),
(6, 0.5193590079534068),
(12, 0.6947976937198777),
(8, 0.6954250622543046),
(5, 0.7063213660937592),
(24, 0.8237595340350533),
(10, 0.8252623244111414),
(18, 0.8724940521021748),
(60, 0.8966469707750959),
(36, 0.8993129699906918),
(30, 0.9084415828523706),
(120, 0.9297237382184487),
(20, 0.9305583483573915),
(48, 0.9332117136873619),
(16, 0.9374475334817786),
(72, 0.955645922168925),
(180, 0.967241667975146),
(9, 0.97065091152207),
(360, 0.9714211873353387),
(240, 0.9774190393054649),
(2520, 0.9869564831528942),
(840, 0.990689138028166),
(84, 0.9942284280532165),
(5040, 0.9944717499826147),
(720, 0.9991264395676062),
(420, 1.0009884030130405),
(1680, 1.007988530141842),
(14, 1.0082283565958516),
(1260, 1.0098444687844825),
(10080, 1.014385392865832),
(55440, 1.017031241946323),
(168, 1.0185506536412827),
(144, 1.0204132800891055),
(27720, 1.0221147099224563),
(7560, 1.0236539502517605),
(15120, 1.0244534419976108),
(42, 1.0273711320299845),
(480, 1.02924063349412),
(96, 1.0301558124571957),
(90, 1.0303059357734239),
(40, 1.0332774802244378),
(1440, 1.03554610947116),
(3360, 1.0361352904051313),
(7, 1.0374988780229564),
(1080, 1.038574163968858),
(30240, 1.0391601774844972),
(20160, 1.0392469286011672),
(25200, 1.040053197716099),
(12600, 1.0418457274652713),
(32760, 1.042601409605724),
(504, 1.0519798537643514),
(50400, 1.052782605677129),
(540, 1.0529251399143578),
(18480, 1.0529349862004858),
(9240, 1.0531920146092764),
(2160, 1.0539968227915038),
(252, 1.05431000605604),
(3780, 1.0563011460182148),
(60480, 1.0597032950898435),
(108, 1.060535543789161),
(13860, 1.062079791253483),
(3960, 1.0621395431253293),
(336, 1.0622338669655944),
(45360, 1.0644912616285656),
(37800, 1.0656024853699857),
(40320, 1.0656787792446576),
(600, 1.0662408035679),
(4200, 1.0664764054509657),
(7920, 1.0669420516567318),
(36960, 1.0671403256721155),
(6720, 1.0681856289274623),
(1800, 1.0682766840513145),
(22680, 1.0689878620794515),
(35280, 1.0707573994918167),
(8400, 1.0709026494840077),
(300, 1.0718010870840695),
(28, 1.0718797484531981),
(42840, 1.0719607117565775),
(1320, 1.073221955060826),
(17640, 1.0741836103228024),
(21840, 1.0744803937858967),
(2880, 1.0745338380025542),
(10920, 1.0756149501471686),
(1008, 1.076857133894279),
(6300, 1.07835204205717),
(216, 1.078353382074721),
(3600, 1.0792095281408574),
(960, 1.0808263306839587),
(4320, 1.0812507096478574),
(47880, 1.0829925384183423),
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Double sharp
Posted: Jan 3 2017, 01:46 PM


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Icarus covered 504 in the "Random Bases" thread, so I did some minor touching up of it to make it a full thread.

Truth be told, though, I think most large bases only need to show the digit map (possibly through an application) with not much of a travelogue, since most people would not seriously consider them as bases. They don't seem to cry out very much for full threads. I made an exception for 504 because it was really easy to copy and paste from Icarus' work and because it is the tetradecimal analogue to 360 (so our heptadactyl civilisation would have "280" degrees to a circle). This makes for nice fractions that look very much like the base-360 fractions of a circle:

{e}
1/2 = 140
1/3 = c0
1/4 = 90
1/6 = 60
1/7 = 52
1/8 = 47
1/9 = 40
1/c = 30
1/10 = 28
1/12 = 23.7
1/14 = 20

But really, I think the highest priorities would be to get bases {2, 3, 4} out. I'd try to do them myself but I presume Icarus has plans for them. Still, I dislike seeing them unfinished when they are the among most important, which I think is the range between 2 and 20 inclusive. Indeed 2 and 3 have a lot to talk about, but is 4 really that much of a problem?

{a}
The interesting thing is that not only do the decimal-friendly {60, 120, 180, 240, 300, 360, 420, 480, 540, 600} look familiar to me, but so do the tetradecimal-friendly {84, 168, 252, 336, 420, 504, 588, 672, 756, 840}. The ones with elevens and thirteens look alien, but somehow the seven doesn't make even the higher {1008, 1176, 1260, 1344, 1512, 1680, 1764, 2016} look that odd. I guess one could interpret it as 7 becoming a normal part of factorisation in the hundreds and thousands, but maybe it's not just that: 54 and 56 look about equally friendly to me in a way that 52 doesn't. Maybe the increased "warmth" coming from 7 as opposed to 11 or 13 is because 7 is a digit and so one is acquainted with its products in the decimal table, and also because it is of about the same scale as 5. But I wonder if tetradecimalists would recognise multiples of b or d in the same way.

I've posted about it elsewhere (so I'll freely borrow my own words), but I tend to mentally imagine the digit map as a corridor of varying length and with no escape (once you go past the last digit, you carry and wrap around to the first). It has lamps hanging at the divisors, and some more poorly illuminating candles flickering at alpha- and omega-related totatives. With a human-scale base, we have to have one or two dark spots (octal 5, decimal 7, duodecimal 5 and 7, tetradecimal 9 and b), but those are few enough in number that we can work around them; we can walk through them, without a need to feel our way around because we should know that region well enough, and come out safely on the other side. (In the case of duodecimal 5 and 7, because there's a strong light at 6, we can simply avoid 5 and 7 whenever possible.)

But when we get to hexadecimal, even though we're mostly in the light for the first half of the digit range, we're blundering around in the dark for most of the top half thanks to the opaque totatives {7, 9, b, d} and opaque semitotative {e}. By the time we get to sexagesimal, even though we have many regular digits that are inviting and usable, there are many stretches where we're reduced to blundering around in the dark (e.g. {41, 42, 43, 44}), because most numbers are not 5-smooth. In base 360, what seems to happen in that extremely long corridor is that we attempt to make a break for it and run from 200 to 216 (the next lamp), and trip over and fall flat on our face around 209.

I think many of the "grand bases" and even the higher natural-scale and mid scale, would tend to need rather shorter tours. I don't think the human mind can actually stay up there and wield pure base 24, let alone 120. We have to break the span and use sub-bases, which is just not the same as real base 24 or base 120, any more than binary is the same as octal or hexadecimal, and is much more complicated and would rather quickly be rejected by the general public (if not ourselves). I think the "human-understandable" range is the one that gets the high-pass coverage and is around two to twenty (I say twenty because it has been civilisational, though only in the past when what was needed of a base was much less strict, and might get defenders from misplaced nationalism or decimal compatibility).

Even further, when push comes to shove, the ones past 15 have a learning horizon problem, the ones below 7 have a number length problem, so while we can stay there longer we eventually need to pull out. It's like trying to sing at the very top or bottom of one's range for a while: it will work for a while (the human voice, just like the human mind, is amazingly tolerant of inefficient usage), but soon it will hurt. Furthermore, using an odd number base seems to be rather alien to our thinking because of the extreme importance of two as a prime. But when we get to {8, 10, 12, 14} part of me thinks that one could not only have a tour, but also an entire constructed world. Consider all the symbolism associated with the decimal digits, after all. How would that change? Would 7 still be a special, sacred number in tetradecimal when it's a divisor? What would take its role? (I suspect b would.) The richness available there is incredible and is, with the exception of decimal (obviously), almost totally unexplored. How would things be different with eight, twelve, or fourteen as a base?

It's an intriguing thought and perhaps comes closest to the idea of having a "tour".
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Oschkar
Posted: Jan 3 2017, 10:56 PM


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QUOTE (Double sharp @ Jan 3 2017, 01:46 PM)
But when we get to {8, 10, 12, 14} part of me thinks that one could not only have a tour, but also an entire constructed world. Consider all the symbolism associated with the decimal digits, after all. How would that change? Would 7 still be a special, sacred number in tetradecimal when it's a divisor? What would take its role? (I suspect b would.) The richness available there is incredible and is, with the exception of decimal (obviously), almost totally unexplored. How would things be different with eight, twelve, or fourteen as a base?

Iíve actually been thinking about this. Notice that the numbers that have the most mystical symbolism associated with them are the smallest opaque coprimes of decimal: 7 and 13. Of course, 7 also gets its sacred quality from the fact that, mostly, only seven Solar System bodies can be seen with the naked eye from Earth, but the fact that there are seven, and not six or eight, seems to have been significant to many ancient cultures.

In an octal conworld, Iíd assume that most of this kind of symbolism will be associated with the "teens". The only single-digit opaque totative of octal is 5. Iím not entirely sure what 5 would correspond to (maybe the planets, excluding the Sun and the Moon?), but there already seems to be a sort of symbolism associated with the pentagram and fivefold symmetry that we donít seem to give much importance to, as a decimal culture. Iíd imagine that in an octal world, the mystical significance of 5 would be seriously emphasized, with 13, 15 and 17 not too far behind.

Duodecimal has both 5 and 7 as opaque totatives, both leaning against the halfway point. Iíd imagine here that 5 and 7 would be seen as opposite in spirit, but exactly what the nature of the opposition is will probably be culturally dependent. Maybe 3 will represent the mundane and 7 the heavenly, with 5 being humanity mediating between them, the head and the limbs adding up to 5.

Tetradecimal has both 5 and 7 transparent; the two opaque totatives are 9 and b. By itself, 9 doesnít seem exclusively mystical because itís 3 sets of 3, but Iím sure that there will be some mythological explanation for its opacity anyway. On the other hand, b is completely inaccessible, and will probably be considered even more mysterious than 7 currently is in decimal.
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Double sharp
Posted: Jan 4 2017, 06:08 AM


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In the West this sort of thing seems to correlate pretty well with the relations of the numbers to the base, while in the East it seems to simply run on what the numbers sound like. (Hence the non-stop tetraphobia which I find incredibly annoying given how omnipresent it is. It is so often and consciously avoided that I've somewhat made it a trolling point IRL ever since I first read about it as a child to include the digit 4 as much as possible whenever it can be chosen for my own things, especially in forms such as 94 "long death", 84 "prosperity [then] death", 74 "death in anger", 14 "easy death", and 64 = 4 * 4 * 4.)

Part of this seems to overlap with more mathematically based reasons, but it can always be taken both ways. In Japan odd numbers are preferred because the evens can be seperated, but in China even numbers are preferred because good things supposedly tend to double. Nine is somewhat equivocal in Japan, because while it is odd, it is also 3*3 (but that usually gets thought of as lucky) and a homophone of "suffering". Somehow eight is lucky despite being even, perhaps because its kanji widens at the bottom (good things multiply, although that seems more like the Chinese version). Four just can't get a break because not only is it even, it also sounds like "death".

I think the first one gives more secure speculation because the second one would require word-construction for the transdecimals. Still, I do think the highest digit would gain some of the symbolism associated with 9 in Chinese culture today (so 7 in octal, b in duodecimal, and d in tetradecimal). In general, though, the homophonous thing may be more relevant for your tetradecimal conlang. happy.gif

Even among the two-digit numbers, it seems that a few get selected and it's hard to tell why. Why 23 and 47 instead of 29 and 41, for instance? Then you have 42 which was selected for looking unremarkable. I cannot begin to imagine how something like this could be ported to tetradecimal, so I think the heptadactyl conworld would need to choose some random numbers from scratch. (EDIT: Perhaps their "answer to the universe" would be an innocuous-looking opaque semitotative product in the line of six as well, so that we might instead get six times eleven, or 4a{e}.)

We would also have different bases being mentioned, perhaps fictionally. Somehow tetradecimal itself is unpopular, but in the heptadactyl conworld I think octal and duodecimal would be common choices (as would perhaps be hexadecimal), with decimal perhaps being a niche and clever choice falling in a sort of uncanny valley of similarity (2*5 vs 2*7). Perhaps the powers-of-two bases would feel "modern" and the highly divisible "c" would feel a little more old-timey, which is the impression I usually get about the views of most people who know a little about this about octal or hexadecimal vs duodecimal (it's a superficial opinion that isn't really true, even though there are legitimate reasons to prefer either eight or twelve as a base). And of course, we'd see octovigesimal vestiges (counting fingers and toes).
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Double sharp
Posted: Jan 5 2017, 08:47 AM


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The classification I'm currently thinking of works like this.

Trivial-scale: {2, 3}. The most fundamental of all number bases. Totally unsuitable for human use in everyday life, but very suitable for machines (hence binary and ternary computing).

Lower natural-scale or small-scale: {4, 5, 6, 7}. Slightly too small to effectively use Stevin's algorithms (because the numbers of digits spiral out of hand), but could be wielded almost as efficiently as the natural-scale bases through tricks like reformulating the multiplicands as sets of instructions and trying to think of two or three digits at a time.

Natural-scale: {8, 9, 10, 11, 12, (13), 14, 15}. (I give tridecimal the benefit of the doubt because at least its magnitude is reasonable, even if the difficulty of its facts is completely unreasonable.) This is the most efficient possible range, the "valley", where you want to settle down and build your civilisation up to today's level of technology. Of these, the even {8, 10, 12, 14} stand out as the most practical bases for a human-like civilisation, although very alien beings might also consider the odd and divisible {9, 15}.

Higher natural-scale or lower mid-scale: {16, 18, 20, 24, 28, 30}. These are the bases which are "too fast to walk, too slow to run". You can't use an alternating-base or reciprocal-divisor system on such bases because it's too slow, but you can't memorise the full tables. (The reason why it's also called the "higher natural-scale" is because the boundaries are fuzzy: the tables of {16, 18, 20} may be reachable by dedicated mnemonists.)

Mid-scale: {32, 36, 40, 42, 48, 50, 54, 56, 60}. These have passed out of the range of thinking of every digit as somehow "equal", but thinking using sub-bases and using complementary divisors or sub-bases work fairly well.

Higher mid-scale or lower large-scale: {64, 70, 72, 80, 84, 90, 96, 100, 108, 112, 120}. Now the complementary divisors themselves and the products you need to remember in the abbreviated multiplication table are starting to become "too fast to walk, too slow to run", and we are starting to need to use sub-bases instead (which works fairly well if you can keep track of both tables, but isn't really using the base; for instance, base 120 is really different from alternating 12 and 10, just like hexadecimal is not binary). These are starting to fall out of the realm of arithmetic and at best may be considered auxiliaries, though the ones up to 84 seem feasible for arithmetic. (That's why the boundaries are fuzzy.)

Large-scale: {144, 160, 168, 180, 192, 196, 210, 216, 240, 252}. This is about the limit of thinking about using two sub-bases. Beyond this, the bases are usable only as auxiliaries.

Higher large-scale: {300, 336, 360, 420, 480, 504}. Essentially, the main reason why you pick a base in this range is to harmonise with the number of days your planet has in a year.

Extra-large-scale: {540, 600, 630, 660, 672, 720, 840, 960}. The higher ones are getting even too fine to use as auxiliaries: once you need to use this kind of precision, you'll probably resort to powers of the base. The exceptions may be things like 672 (the tetradecimal version of 480) and 660 (which essentially is the only good choice for an undecimal auxiliary base).

Grand-scale: Everything else. Among lower grand bases we have things like {17, 19, 22, 23, 25, 26, 27, 29, 31}, which are in the same scale as the higher natural-scale but don't have any factors to help them and just fall out of usability. Among higher grand bases you have numbers like {1080, 1260, 1440, 1680, 2160, 2520, 3360, 3780, 3960, 4200, 4320, 4620, 4680, 5040}, where the number of factors is amazing, but the numbers are just too great and require too much effort to use even as auxiliaries (since they won't harmonise well with any natural-scale base).
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Double sharp
Posted: Jan 31 2017, 08:20 AM


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@Icarus: would you mind adding my new "guest posts" for bases 32, 54 and 504 to the tour menu thread?

The ones I'm most interested to see your posts for at the moment are {22, 26, 27}, which seem to want the most engaging travelogues. I would also once again suggest {168, 180} as two of the most important grand bases (though some of Oschkar's suggestions are also very interesting.)
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Double sharp
Posted: Feb 5 2017, 04:50 PM


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What Bases Might be Included Past the Human Scale?

Looking at what bases (except for prime detectives) are already included in the tour, it seems to me that up to 36, we need at least five divisors:

Five divisors: 16 (tess)
Six divisors: 18 (dine), 20 (score), 28 (cadeff), 32 (twive)
Eight divisors: 24 (cadex), 30 (kinex)
Nine divisors: 36 (exent)

And up to 64 we need at least seven (I'm planning a guest post for 64):

Seven divisors: 64 (twex)
Eight divisors: 40 (kinoct), 42 (exeff), 54 (exove), 56 (sevoct)
Ten divisors: 48 (exoct)
Twelve divisors: 60 (shock)

And then up to 120 we need at least nine:

Nine divisors: 100 (kent)
Ten divisors: 80 (octess), 112 (setess)
Twelve divisors: 72 (octove), 84 (sezzen), 90 (novess), 96 (extess), 108 (catrine)
Sixteen divisors: 120 (hund)

There have then been so many numbers with 12 divisors that to continue, we seem to need at least fifteen up to 240:

Fifteen divisors: 144 (zenent)
Sixteen divisors: 168 (cadexeff), 210 (zeffchick), 216 (excue)
Eighteen divisors: 180 (scorove)
Twenty divisors: 240 (kinexoct)

and at least eighteen past that:

Eighteen divisors: 252 (zentress), 288 (notwive), 300 (zenquint)
Twenty divisors: 336 (exsevoct)
Twenty-four divisors: 360 (kinoctove)

and 360 is so hard to beat that in the range past that we really need to match it:

Twenty-four divisors: 420 (exsevess), 480 (exodess), 504 (sevoctove), 540 (exovess), 600 (cadexquint), 630 (senovess), 660 (exdessell), 672 (exeftess)

before we finally beat it.

Twenty-eight divisors: 960 (zenodess)
Thirty divisors: 720 (octovess), 1008 (senotess)
Thirty-two divisors: 840 (seveszen), 1080 (noveszen)

And now, only the actual largely composites seem to stay interesting, as the lanterns in Oschkar's staircase flicker out and leave us groping in the dark.

Thirty-six divisors: 1260 (scorsenove), 1440 (novestess)
Forty divisors: 1680 (sechitess), 2160 (kintestrine)

And finally, all the companions of kinsevoctove:

Forty-eight divisors: 2520 (kinsevoctove), 3360 (sechitwive), 3780 (scorsetrine), 3960 (kinoctovell), 4200 (exsekent), 4320 (kintwivetrine), 4620 (exsevessell), 4680 (kinoctovithe)

before we exit to the roof of the skyscraper.

Sixty divisors: 5040 (sevoctovess)

All these are just "gut feelings", of course, fuelled by possibly inappropriate extrapolation.
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icarus
Posted: Aug 15 2017, 08:35 PM


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Work has resumed on the coding effort. See the update log post.
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Double sharp
Posted: Oct 25 2017, 03:44 AM


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I have been going through some of the older threads and "upgraded" them to include some of the data that typically goes in, such as full multiplication tables to base 30, abbreviated tables, and so forth. Next will probably be the addition of abbreviated tables to the bases {21, 35}, and maybe {17, 19} even though the abbreviation for these primes is meaningless, because it is done for {109}.

Regarding plans for future tours, I will probably make {45, 50, 64} soon; the last one brings us another binary power, 45 brings us an odd 5-smooth base of the form α2β to see how they look, and the middle one is one of Treisaran's "near misses" of the form αβ2. This would complete the decimal multiples as we have a complete set of dozenal multiples up to ten.

Meanwhile, I have asked Icarus to generate {22, 26, 27}, since he previously remarked that he had notes for these; he has some time this week, so they should be coming. I think {2, 3, 4} will remain unlinked for now. (Nevertheless, I wonder if I could do them. I get that binary and ternary require a lot of talk about their applications, but surely base 4 is no harder to cover in a tour than base 6?)

I have added a few abbreviations of the tables for 16, but given that this base is likely in the human-scale, it starts to become a little suspect if this makes sense. I think it does because rolling odd products is essentially the only way to learn the difficult totative lines, although the line of c may not be that hard and may be taught for itself as well as by rolling the threes.

Icarus' sketchbook contains abbreviated tables for the bases {10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 40, 42, 44, 45, 48, 54, 56, 60, 64, 66, 70, 72, 80, 84, 90, 96, 120}. I think this is a little irrelevant for the human-scale bases, so I will just use this as a guide for the bases not in anyone's plans for the tour yet: {33, 38, 44, 66}. I am not sure if these clunky products of 11 and 19 with few redeeming features are going to interest anyone, but if anyone cares, you can reply and I'll put them in my plans...

As for the grand bases, I don't think we need any more detailed tours for the most part, except maybe for {168, 180, 216, 420, 840}. What I think we could do is have a "grand base mashup", which would give digit and divisibility maps, together with vital statistics, of {120, 144, 168, 180, 210, 216, 240, 252, 288, 300, 336, 360, 420, 480, 504, 540, 600, 630, 660, 672, 720, 840, 960, 1008, 1080, 1260, 1440, 1680, 2160, 2310, 2520} in its most ideal form. I think this is easy to automatically generate with no accompanying verbal explanation but headers like "Base 300 = 2^2 * 3 * 5^2", but some of the larger ones might not fit in one post. If they somehow do, then maybe we can think about the leviathans {3360, 3780, 3960, 4200, 4320, 4620, 4860, 5040} wink.gif, though I believe the point has been made by the time we hit 2520.
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Oschkar
Posted: Oct 25 2017, 05:46 AM


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Iím willing to do {168, 180}, but I donít really have the time for it...

Do we have a list of octal and tetradecimal multiples up to ten?

{8, 16, 24, 32, 40, 48, 56, 72, 80} are already covered, and {64} is within your plans.

{14, 28, 42, 56, 70, 84, 112} are covered.
{98} doesnít seem useful; itís another of Treisaranís αβ2 near misses, but one that is worse both intrinsically and extrinsically than {50}, and at twice the size.
{126} may seem interesting just for being a product of 2, 3 and 7 next to a power of 5, but itís only singly even.
{140}, on the other hand, doesnít seem to have any redeeming features. Its prime signature is α2βγ, like that of {60, 84}, but as icarus said in the title for base 28, ďtoo much water weakens the teaĒ. (However, we can imagine a society using base 28, at least at a stage at which most people donít need to do much math beyond counting and adding.)
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Double sharp
Posted: Oct 25 2017, 07:02 AM


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Hmm, yeah, I should add {126} to my list. It is kind of like 80, being next to a perfect power of the prime it misses in its factorisation. 10 has the same pattern at a smaller level, and if we relax things a little {24, 26, 28} also show up as possibilities. The only problem is that 126 is only singly even, as you say. (And the next even is the septuply even 128! But neither 128 nor 256 have anything really interesting in their neighbours, so I'll give them a miss.)

Maybe 105 = 3 ◊ 5 ◊ 7 deserves a tour, too. Or even the abundant 945 = 3^3 ◊ 5 ◊ 7.

I think Icarus' already covered 99 already shows all the good stuff about 98 and unites it with the squares from the decimal hundred. (And I have a hard time not calling 98 and 99 "duodecentesimal" and "undecentesimal" respectively. biggrin.gif) Fifty has a better neighbour, although I feel sorry for 2 and 3 in it, being upstaged by 5 and 7. happy.gif

I think most bases covered here, except the really grand ones, could be used civilisationally at the stage you mention. I'd say that that's the level it makes sense to think about all those more baroque means of computation, and posit a lower mid-scale from 18 to 60 and a higher mid-scale from 60 to 120. But only for hexadecimal does it ever seem to get as good as a human-scale base, so much so that I would call it one.

Among hexadecimal multiples, {128} is pretty useless, and {160} is another case of "too much water weakens the tea".

Mashups

I'm intrigued by the ones we already have (mid-scale, 60 vs 120, dozens, primorials, prime detectives), so I'll suggest a few more:

1. Power of 2 mashup (up to 256 or 512, mentioned earlier in the thread)

Every number is next to a multiple of 2 or 3, if it's not 1; every square to a multiple of 5; every cube to a multiple of 7; every fifth power to a multiple of 11; and so on (just subtract 1 from the prime and halve it). The powers of 2 are the smallest numbers of the form α^n that can illustrate this.

2. Prime mashup (up to 19? 31?)

Because λ(24) = 2, and λ(n) > 2 for all n > 24 (where λ is Carmichael's reduced totient function), every prime square from 5^2 onwards is one more than a multiple of 24. This all but guarantees good omegas, and sometimes the alphas are also interesting: at the very least, one of them must also contain 5, because λ(240) = 4.

5^2: α = 2 * 13, ω = 2^3 * 3
7^2: α = 2 * 5^2; ω = 2^4 * 3
11^2: α = 2 * 61, ω = 2^3 * 3 * 5
13^2: α = 2 * 5 * 17, ω = 2^3 * 3 * 7
17^2: α = 2 * 5 * 29, ω = 2^5 * 3^2
19^2: α = 2 * 181, ω = 2^3 * 3^2 * 5
23^2: α = 2 * 5 * 53, ω = 2^4 * 3 * 11
29^2: α = 2 * 421, ω = 2^3 * 3 * 5 * 7
31^2: α = 2 * 13 * 37, ω = 2^6 * 3 * 5

3. SHCN mashup

Much more practical than the HCN mashup, going to 2520 or 5040 with a similar range to the primorials.

4. Grand base mashup

Already explained in my previous post.

This makes for very good indirect relationships that don't quite assuage the problem of severe detuning past 7 or maybe 13.

5. [2, 1] prime signature mashup (edited in later)

Mentioned in Icarus' tour post for {18}, and I think it would be interesting to compare the human-scale {12} with the higher natural-scale bases, like the almost-manageable {18, 20} and the more unwieldy {28}, and then the totally oversized {44, 45, 50, 52}.

Side Reading Material

I'll shamelessly plug my human-scale and your higher natural-scale threads here. I'd have plugged the lower natural-scale one, but only 6 really stands out there, and I have treated it as a human-scale base throughout (like 16).

One thing that seems a bit lacking is a discussion of the "odd natural scale". It seems to me that the odd bases {7, 9, 11, 13, 15} need significant pedagogical changes from the even natural scale, and I am curious if the fact that I can't think of solutions for them is anything more than just a lack of imagination.

The odd human-scale bases have a prime power (9), a semiprime (15), and some primes (7 as a medium one and 11 and 13 as fairly large). (I think 5 is stuck between a rock and a hard place, and if 18 and 20 are non-trivially too hard, then 17 and 19 must be write-offs.) What they lack is a vast composite like 12, which no odd numbers match until 45.
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Double sharp
Posted: Nov 2 2017, 02:04 PM


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Base 180 is here! happy.gif

I've noticed that alternation is a much more extendable method for bringing lots of "weird numbers" in the lower mid-scale into personal use, mostly because complementary divisors and other "single-digit methods" seem to get exhausted unless your base really is largely composite. 24 and 30 work well with complementary divisors, but 28 doesn't really: it seems to need to be treated as {2:14} or {4:7}. And while stuff like {21} or {35} have severe tuning problems in complementary divisors thanks to 20 and 36 being next door respectively, they become thinkable as {3:7} and {5:7}.

Most of these semiprimes look rather boring, but one caught my eye in particular: {65}, which is 5*13 and neighbours 64 = 2^6 (granting it immense binary-power resolution for an odd base) and 66 = 2*3*11 (granting it immense infill for most of the primes it lacks). It is quite like the "prime detectives" {21, 34, 55, 99} in this respect.

P.S. 1200 posts! happy.gif
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Double sharp
Posted: Nov 3 2017, 02:33 PM


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Base 64 has been completed, and {480, 960} are in the works based on Icarus' other post for 960 for the ginormous digit map that these entail. Hopefully I can get 45 and 50 posted up soon. Together with {22, 26, 27} coming from Icarus, that should easily give you all a good deal of reading material for the start of November! happy.gif
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Double sharp
Posted: Nov 4 2017, 09:07 AM


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And here is 126, as noted by Oschkar. I hope the sin of not including the AMT is forgivable, because the one for 120 is already borderline too big, and it seemed like too much work for not much gain. happy.gif
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Oschkar
Posted: Nov 5 2017, 05:07 AM


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So, Iíve been looking at Robinís inequality once more, since my new computer was able to compute the ratio for all numbers up to 21621600 in less than half an hour.

I limited the output to the bases with a ratio under 9/8, just to keep the amount of data manageable. There are 800 such bases, the smallest of which are eminently recognizable to us. However, since the function depends only on the divisors, excluding any influence from the neighbour-factors, itís difficult to obtain a completely fair assessment of the utility of many of these numbers as bases. Regardless, it is very close.

For example, {56, 80} donít quite make the cut, whereas {132} does. Although 132 does have more and larger divisors than both 56 and 80, it is less suited to be a number base. It contains the prime factor 11, which is too large for subitization, and doesnít have the neighbour relationships that a base in its scale should have.

On the list, there are numbers with up to 10 binary powers, 6 ternary powers, 4 quinary powers, 3 septimal powers, 2 powers each of {11, 13, 17}, and 1 power of the following primes up to 47. The largest binary-power, 3-smooth, 5-smooth and 7-smooth bases are {32, 432, 10800, 1209600}, respectively. On the other hand, the largest bases in the list not divisible by {2, 3, 5, 7} are {15, 40, 2016, 617760}, respectively.

Discarding non-7-smooth numbers and the subitizable bases {2, 3, 4}, the smallest bases not yet included in the Tour are currently {168}, the tetradecimal long-hundred analogue; {192}, a byte three-quarters full; {216}, the senary thousand; {252}, the tetradecimal half-turn, and {288}, which is two gross to even begin to describe.

Attached File ( Number of downloads: 18 )
Attached File  robin.txt (22.69 kb)
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Double sharp
Posted: Nov 5 2017, 05:37 AM


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I see you've been reading some of my hopeless speculation about subitising against {2, 3, 4, (5)} and primes over 7... happy.gif

To be fair, the neighbours will provide much less penetration than the divisors. This seems obvious: if your number has a lot of divisors, then bumping it up or down by one will turn them all into coprimes. Thus it's not really surprising why so many HCNs are primeflanks.

Even in something like 120, 119 and 121 are surely much less divisible than 120. Their importance comes from the fact that they have the smallest coprimes: 7, 11, and 17 provide significantly more penetration than 59 and 61.

In a way, levanunqual (base 132) is already covered in the dozens mashup. I'm also not terribly interested in it, but I am intrigued by the idea of adding 66 to the tours. For one thing, while it is clear that switching 7 for 11 is not a good deal, surely we need to show the consequences once, by taking the good pattern of 2*3*5=30 and 2*3*7=42 and continuing it to the not-so-desirable 2*3*11=66. (For similar reasons, I think there's a place for 3*5*7=105.)

Some of these numbers are echoes of the power of 24, the double dozen and crown jewel of the lower mid-scale: it controls its totatives and their squares perfectly, so that all totatives square to 1 mod 24. This is why, next to many of the prime squares, we get nice dozenal multiples: {24, 48, 120, 168, 288, 360} are all highly divisible numbers that are nicely positioned, and many of the small primes seem "better" than they really are as bases thanks to them. (528 isn't nearly as nice, probably because of that eleven-factor, but 840, 960, 1680, and 5040 are interesting.)

Most of these are pretty anonymous, though. By the time we get to 240, we already have a fairly bad load of semitotatives, and the HCNs from 360 onwards all start to look alike. The tetradecimal analogues {168, 336, 504, 672} are perhaps the most interesting of the bunch. But then again, that makes them pretty easy to "phone in", so that you might actually see some of these leviathans before I get around to 50...
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Oschkar
Posted: Nov 5 2017, 05:55 AM


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What I find interesting about the sequence is its sub-exponential growth. I donít understand genetic programming yet, so I canít write a symbolic regression script to determine the best fit function; however, it was too steep for power regression and too slow for exponential regression.
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Double sharp
Posted: Nov 5 2017, 06:09 AM


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QUOTE (Oschkar @ Nov 5 2017, 05:55 AM)
What I find interesting about the sequence is its sub-exponential growth. I donít understand genetic programming yet, so I canít write a symbolic regression script to determine the best fit function; however, it was too steep for power regression and too slow for exponential regression.

Here is base 480, born out of copy-pasting from 960, and procrastinating for 168.

It would be interesting to see what happens to the ratios in the very long run. They really do dip significantly at the SHCNs: 55440 is much lower at 1.01 than all the others around it near 1.1, and a similar effect happens at 720720, 1441440, 4324320, and 21621600. Theorem 323 of Hardy and Wright says that lim sup σ(n) / n log log n = e^γ and it would be nice to see this in action - that is, if the scale of the numbers involved was not too impractical. happy.gif

P.S. Having done {480, 504}, it seems that I have more or less forced myself to include all the "companions of kinoctove" as well in the LCN sequence. So, we have to do {168, 336, 420, 540, 600, 630, 660, 672}, and then also {216, 252, 300, 840}. Well, you can probably expect to see some of them soon...
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Oschkar
Posted: Nov 5 2017, 06:22 AM


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Well, Iíll try, but my algorithm seems to run in something like O(n2 log2 n) time. Perhaps I should limit myself to SHCNs, which, having relatively small prime factors, wouldnít strain the prime-generating function for Eratosthenesí sieve as much.
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