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This was meant to be 2 and d, right? Earlier, your script writes "d if the tetradecimal digital root of x is divisible by d", so I'm assuming that all these numbers are supposed to be expressed in whatever base is being examined at the moment, not decimal.icarus @ Oct 31 2015, 02:08 PM wrote:
I think it would be mighty interesting to see lists of sexagesimal and centovigesimal divisibility tests. However, you raise a good point about stating ranges. Maybe at some point you'd shift to simply giving the rule without stating what digits are involved? For example, the rule for 2 in pure sexagesimal is that a number is even if it ends in one of {0, 2, 4, 6, 8, a, c, e, g, i, k, m, o, q, s, u, w, y, A, C, E, G, I, K, M, O, Q, S, U, W}. Hmm, quite a mouthful. But what if we just said "if the least significant place value of x is divisible by 2"? That would get the point of the test across, without drowning the reader in so many even digits. (Although maybe you do want to illustrate that point where even the divisor tests can reach exhaustion!)icarus @ Nov 4 2015, 11:02 PM wrote:Double Sharp,
Yes! There are some validation items that need to happen. I do need to ensure that any figure in the prose is stated in base b EXCEPT parenthetical ones, which are decimal. (There will be a note to this effect).
I am holding on fixing the script to represent large bases (here meaning bases greater than 36, for no other reason than I have to append the next sequence of transdecimals beyond lowercase letters). Do I want to *still* be putting out these sorts of verbal descriptions, or just rely on the digit map? I lean toward YES. If that's so, then do we continue the base-b representation or cut it off? This effects range-folding and the ranges stated for regulars. At some point we run out of transdecimals. I think at some point I don't give ranges / range-folding but merely state that there are so many combinations. For bases greater than 60 divisor tests become cumbersome; stating ranges is going to suck up a lot of real estate. Going to need to ponder it, what are you's all's thoughts?
I think any base with a gapless prime factorisation like {2, 3} has a real problem with auxiliary bases. To get the factor 5, it goes into the vicious cycle I've mentioned several times already:icarus @ Nov 4 2015, 11:02 PM wrote:Today I worked on the "segment A" again. Pretty satisfied with it so far.
Segment A includes the names, prime decomposition, prime signatures, classes in which base b resides. I think it's nailed. The LaMadrid and SDN names are given, then the prime decomposition and prime signature. The script produces a brief range of numbers with similar prime signature and declares which OEIS number and if available, classes such as "sphenic numbers", "squarefree semiprimes", etc. I have yet to add "primorials" and other types of number independent of prime signature. Want to add primorials, HCN, SHCN, highly regular number, pronic number, etc. and will do that. Here are samples:
The prime cases should perhaps more simply read "p is prime, its prime signature is "1". The number p is term _primePi(p)_ in the sequence of primes is {2, 3, 5, ..., p, ...} (OEIS A000040)." This can be handled by an If statement that caters to primes. I think anything else can be treated in a more or less uniform way. [DONE 201511041718]
Also, hyperlinks to definitions need to be supplied as are in the tour. There are two separate references, one here and one at the home to-be. This will be added later. I think the hyperlinks are crucial so won't leave them out.
The OEIS hyperlinks can easily be made clickable because the site is so modular. (In fact I will do that now). [DONE 201511041718]
Next week I could lose focus due to business (At this point I would welcome that) but will resume when free time opens up again (feast or famine self employed).
I might tackle the generation of auxiliary bases if that is even possible. Will have to define criteria. Resolution of missing or ensuing primes is easy; what to do for bases that don't "need" it? This will favor resolution of the first three primes {2, 3, 5}, with preferred multiplicities {3, 1, 1} but this might require much thought. Again, what are your thoughts?
Remember to post base 54 in "Senary and {2, 3} bases" as 54 is 2 3Â³.Double sharp @ Oct 17 2015, 11:52 AM wrote: My intent would be to cover {40, 54, 56} if nothing else. Following that, I'd go to {44, 45, 50, 52}, and then the binary powers {32, 64}.
Huh. So I see in the hidden comments to this thread the following planned tours: {2, 3, 4, 22, 26, 27, 32, 35, 45, 54, 420, 840, 1000, 1260, 1680, 1728, 2310, 2520, 5040}.icarus @ May 1 2012, 03:04 PM wrote:
I'm going to suggest 42 and 84 together with the initial mid-scale and 9 and 15 as the human-scale composite odd bases.icarus @ Jan 18 2016, 11:35 PM wrote: If you have early suggestions I'd appreciate them.
The first bases will be: {6, 8, 10, 12, 14, 16}, {20, 24, 30, 36, 60, 120}, {210, 2310}, {5, 7, 11}, {21, 34, 55, 56, 64, 76, 99}.
Going to add {1680} to the huge bases. I'm fairly sure that I can work in 12:10:14 a bit more easily than in 12:15:14, with all three subbases being even, but we lose the nice omega relationship with 11, which then becomes a maximal prime.Double sharp @ Jan 19 2016, 08:13 AM wrote: Please don't forget {240, 360} and the huge {2520, 5040}! I also think 54 deserves a place in there.
In the full map, we also need colours for compound square-alpha tests, like pentaquinquagesimal 51 (Î±Â²Ã—Ï‰), 68 (Î±Â²Ã—Î±) and 272 (Î±Â²Ã—Ï‰Â²). I'm going to suggest for these a medium blue-violet, about #865edb, and for things like pentaquinquagesimal 255 and 340 where they combine with the divisors a lavender like #da89fa for richness 1 and #e4a7fc for higher richness.
Youâ€™re probably right that compound square-alpha tests are probably not something that could be reasonably applied in most bases, especially ones like 55, but they might turn out to be intuitive in some small bases, say, testing for 15 in octal, where 5 comes from the square-alpha and 3 from the alpha itself. I donâ€™t think that the three types of compound square-alpha tests need to be distinguished in the charts, though.icarus @ Jan 20 2016, 12:08 AM wrote: One of the problems we might run into is, where do we draw the line in saying a test is "intuitive" or not? Now I think I have the ability to "shut off" items that are more obscure using bits. I would enable such tests using the fifth bit. If I "run out of bits" I can "borrow" the seventh (that which enables the regular figure categorization) since it nullifies the effect of most of the other bits. I could use a Boolean method to enable specialty tests to show.
{a}icarus @ Jan 20 2016, 12:47 PM wrote: Indeed omega-2 is not very useful but was a by-product of panning for semicoprimes (products of a regular g and coprime t) that have a coprime part t that is a factor of (b^4 - 1). Now that I might pre-sort the numbers n in the range under study (often that of b itself) the need to handle omega-2 is gone. I could color omega 2 "hh" or "opaque semicoprime". This would free up a "slot" in HTML.