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icarus 
Posted: Oct 1 2013, 12:15 PM


Dozens Demigod Group: Admin Posts: 1,913 Member No.: 50 Joined: 11April 06 
Base 85  Pentoctagesimal (Dozenal: Base 71;  Septunimal) The integer 85 is a semiprime with nonconsecutive prime factors {5, 17}. Decimal is the smallest example of such a semiprime. As a semiprime, base 85 has 4 divisor digits {0, 1, 5, 17}. The smaller divisor squares below the base, thus base 85 has a single regular digit (semidivisor) 25. Its totient ratio is (4/5 × 16/17) = 64/85 = 75.29% (90;51 p/g., 90:42 twelftieths). The pentoctagesimal digits that are not divisible by either {5, 17} are coprime to 85. The ωnumber is 84 (dozenal 70) with a dozen factors {1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84}. The αnumber is the semiprime 86 (dozenal 72 with factors {1, 2, 43, 86}. We have intuitive divisibility tests for the four smallest primes {2, 3, 5, 7} and of course 17. Many multiples of {5, 17} are transparent. This base is very noticeably "out of tune". With the exception of the adjacency of pentoctagesimal to the highly divisible 84, the base is a typical semiprime, otherwise unremarkable. Digit Map
Intuitive Divisibility Tests
Regular Digits


icarus 
Posted: Oct 1 2013, 12:58 PM

Dozens Demigod Group: Admin Posts: 1,913 Member No.: 50 Joined: 11April 06 
Anirudhk,
I looked up what I studied and it appears I only *think* 82 popped up. The study in question regards residues of the geometric progression of 2 in base 360. That sequence runs {1, 2, 4, > 8, 16, 32, 64, 128, 256, 152, 304, 248, 272, 184}, repeating at the arrow infinitely thereafter. The terms beyond 256 are all at least 8 times a prime, {19, 31, 17, 23} respectively. So there are a bunch of semirandom bases to study in that weird sequence. 
icarus 
Posted: Oct 1 2013, 12:59 PM


Dozens Demigod Group: Admin Posts: 1,913 Member No.: 50 Joined: 11April 06 
Base 27  Heptavigesimal (Dozenal: Base 23;  bitrimal) The integer 27 is the cube of three. As a prime power p^{ε}, it has (ε + 1) divisors. Thus heptavigesimal has 4 divisor digits {0, 1, 3, 9}. Its totient ratio is that of 3, i.e., 2/3 = 66.67% (80 p/g., 80 twelftieths). If we use nonary coded heptavigesimal as a template, we can effiiently map the larger base’s digits in a compact way, as these inherit the smaller base’s totient pattern. The heptavigesimal digits that are divisible by three are not coprime to 27. Heptavigesimal nondivisor regular numbers must all exceed 27, since it is a prime power with one distinct prime factor. Any positive integer power of 3 less than 3³ must divide 27, so all regular digits are also divisors. All the positive integer powers of three are regular in base 27. The ωnumber is 26 (dozenal 22) with factors {1, 2, 13, 26}. The αnumber is 28 (dozenal 24) with factors {1, 2, 4, 7, 14, 28}. Thus we can use the omega divisibility test on the primes {2, 13} and the alpha on primes {2, 7}. There are a fair number of small digits that enjoy heptavigesimal intuitive divisibility tests. As an odd base, it has the minimum indirect coverage of the powers of 2, i.e., 2 as both alpha and omega, and 4 as alpha. Digit Map
Intuitive Divisibility Rules
Regular Digits


icarus 
Posted: Oct 2 2013, 11:44 AM


Dozens Demigod Group: Admin Posts: 1,913 Member No.: 50 Joined: 11April 06 
Base 133  Centotritrigesimal (Dozenal: Base E1;  Levunimal; “Base Lee”) The semiprime r = 133 is the product 7 and 19. It has 4 divisor digits {0, 1, 7, 19}. Its totient ratio φ(r)/r is 108/133 = 81.2% (98;E2 p/g., 97:53 twelftieths). The ωnumber is the highly divisible 132 (dozenal E0;). The αnumber is the semiprime 134 (dozenal E2;) = {2, 67}. Thus conceptually, we can use compound divisibility tests on semicoprime numbers that are products of any positive power of the prime divisors {7, 19} and {2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 67, 132, 134}. This base resembles base 85. Aside from its proximity to (11 × 12) that lends it many intuitive divisibility tests, there is little remarkable about the Dixie base. Digit Map
Intuitive Divisibility Tests
Regular Digits


dgoodmaniii 
Posted: Oct 2 2013, 01:05 PM


Dozens Demigod Group: Admin Posts: 1,927 Member No.: 554 Joined: 21May 09 
Levunimal, actually. But "Base Lee" still works. 

anirudhk 
Posted: Oct 2 2013, 01:08 PM

Casual Member Group: Members Posts: 105 Member No.: 739 Joined: 8August 13 
Base 770 (542 in dozenal)
2 * 5 * 7 * 11 This base reminds me of spearmint, I have no clue why. Pros It's divisible by many small primes. It's the first number not divisible by 3 to have a totient ratio lower than that of 3 smooth numbers. (31%) It's not a prime flank (Of course, all numbers which aren't divisible by 3 are ). Cons It's not divisible by 3. Though it's even, it's not divisible by even 4. 3 shares an alpha relationship, which is the worst relationship with 3. I would rate this base 2.1/6 stars. 
icarus 
Posted: Oct 2 2013, 05:28 PM


Dozens Demigod Group: Admin Posts: 1,913 Member No.: 50 Joined: 11April 06 
Base 504  Quincentoquadral (Dozenal: Base 360;  trihexanunqual) The integer r = 504 is the product of 8 primes, three of which are distinct, with the prime decomposition {2³, 3², 7}. It has 24 divisor digits {1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 18, 21, 24, 28, 36, 42, 56, 63, 72, 84, 126, 168, 252, 504}. The quincentoquadral divisor ratio σ_{0}(r)/r = 1/21 = 4.76% (6;X3 p/g, 5:86 twelftieths). Its totient ratio φ(r)/r is that of 42 = {2, 3, 7}, i.e., 2/7 = 28.57% (35.19 p/g., 34:34 twelftieths). The quincentoquadral digits that are {1, 5} mod 6 and not multiples of 7 are coprime to 504. The ωnumber is the prime 503 (dozenal 35E). The αnumber is the semiprime 505 (dozenal 361). Thus conceptually, we can use compound divisibility tests on semicoprime numbers that are products of any positive power of the prime divisors {2, 3, 7} and {5, 101, 503}, the latter two of the latter set not too useful. A solid block of quincentoquadral intuitive divisibility tests exist for the first ten digits. This number swaps the factor 5 with 7 in the prime decomposition of 360. It is a factor of 2520. Digit Map


icarus 
Posted: Oct 2 2013, 05:32 PM


Dozens Demigod Group: Admin Posts: 1,913 Member No.: 50 Joined: 11April 06 
[more 504] Intuitive Divisibility Rules
Regular Digits


wendy.krieger 
Posted: Oct 3 2013, 07:24 AM

Dozens Demigod Group: Members Posts: 2,432 Member No.: 655 Joined: 11July 12 
Base 504 was one of the bases considered here. This was a program run to find the size of regulars against their 'coregulars' (ie recriprocals). Base 504 has a very large plateau and sharp sides, due to 7, 8, 9 being pretty much the same size.

icarus 
Posted: Oct 3 2013, 11:29 AM


Dozens Demigod Group: Admin Posts: 1,913 Member No.: 50 Joined: 11April 06 
Base 29  Enneavigesimal (Dozenal: Base 25;  bipentimal) The prime base 29 is limited to the 2 trivial divisor digits {0, 1}. Its totient ratio is 1/p, i.e., 1/29 = 3.45% (4;b7 p/g., 4:17 twelftieths). A prime base cannot have neutral digits; all digits must either divide or be coprime to the base. Every digit that is not {0, 1} therefore must be coprime. Only the positive integer powers of 29 itself are regular in base 29. The ωnumber is 28 (dozenal 24) with factors {1, 2, 4, 7, 14, 28}. The αnumber is 30 (dozenal 26) with factors {1, 2, 3, 5, 6, 10, 15, 30}. Thus we can use the omega divisibility test on the primes {2, 7} and the alpha on primes {2, 3, 5}. Despite the monotony of the digits, base 29 does enjoy fortuitous neighbor relationships. It has divisibility tests for the smallest four primes, with some deeper power to scan for two {2², 3, 5, 7}. Digit Map
Intuitive Divisibility Rules
Regular Digits


icarus 
Posted: Oct 3 2013, 07:06 PM


Dozens Demigod Group: Admin Posts: 1,913 Member No.: 50 Joined: 11April 06 
Base 119  Centenneadecimal (Dozenal: Base 9E;  ennlevimal) The integer 119 is a semiprime {7, 17} adjacent to a colossally abundant number. Thus it has 4 divisor digits {0, 1, 7, 17}. Its totient ratio is (6/7 × 16/17) = 96/119 = 80.67% (98;20 p/g., 96:97 twelftieths). The centenneadecimal digits that are not multiples of at least one of the prime divisors {7, 17} are coprime to 126. The ωnumber is 118 (dozenal 9X) with factors {2, 59}. The αnumber is the colossally abundant number 120 (dozenal X0) = {2³, 3, 5}. Thus conceptually, we can use the omega compound divisibility test on semicoprime numbers that are products of any positive power of the prime divisors {7, 17} and one of the powers 0 < τ ≤ 3 of t_{α} = 2 and the other neighbor factors {3, 5, 59}. Because of the many neighbor factors, base 119 hase many intuitive divisibility tests for small integers. This semiprime is only remarkable because of its adjacency to 120, otherwise it plays much like bases 85 and 133. This number has a severe tuning problem that we can adjust by tuning up to base 120. Digit Map
Intuitive Divisibility Rules
Regular Digits


dgoodmaniii 
Posted: Oct 3 2013, 08:23 PM


Dozens Demigod Group: Admin Posts: 1,927 Member No.: 554 Joined: 21May 09 
Ennlevimal; we expect "nov," but that initial "n" would conflict with "nil," so we use the Greek root "enn" instead. I probably sound hypercritical, since my only two messages have been SDN pedantry, but I'm enjoying this and hope you all continue doing it. 

icarus 
Posted: Oct 3 2013, 09:07 PM

Dozens Demigod Group: Admin Posts: 1,913 Member No.: 50 Joined: 11April 06 
Don I purposely put the wrong SDN up there so I can gauge if anyone's watching. jk. No, and I am wrong about all the novo, its ennea. You can see the reason why I don't make a good corporate wonk. I simply don't try well enough to follow rulez. "I ain't a team playya". Thank the Lord for the edit button!

icarus 
Posted: Oct 3 2013, 09:12 PM


Dozens Demigod Group: Admin Posts: 1,913 Member No.: 50 Joined: 11April 06 
Base 40  Quadragesimal (Dozenal: Base 34;) The integer 40 is the product of four primes, two of these distinct, {2³, 5}. It has 8 divisor digits {0, 1, 2, 4, 5, 8, 10, 20}. Its totient ratio is that of 10 = {2, 5}, i.e., 2/5 = 40% (49;72 p/g., 48 twelftieths). The quadragesimal digits that are odd but not divisible by 5 are coprime to 40. The ωnumber is 39 (dozenal 33) with factors {1, 3, 13, 39}. The αnumber is a prime, 41 (dozenal 35). Thus we can use the omega divisibility test on the primes {3, 13}. Despite the greater accommodation for powers of two, there doesn't seem to be an advantage to turning to base 40 over decimal. The properties of decimal, especially the indirect relationships with {3, 9, 11} are easily more useful than {3, 13, 39, 41}, all in a much more compact base. Digit Map
Intuitive Divisibility Tests
Regular Digits


Oschkar 
Posted: Oct 4 2013, 01:45 AM


Dozens Disciple Group: Members Posts: 575 Member No.: 623 Joined: 19November 11 
Base 49  Enneaquadragesimal (Dozenal: Base 41;  quadrunimal) The integer 49 is the square of 7. Because it is the square of a prime, it has only three divisor digits, {0, 1, 7}. Its totient ratio is 6/7 = 85.71%, and any digit not divisible by 7 is coprime to 49. However, 49 has very useful neighbours. The ωnumber is 48, a highly composite number with ten factors {1, 2, 3, 4, 6, 8, 12, 16, 24, 48}, and the αnumber 50 has six factors {1, 2, 5, 10, 25, 50}. These factors combine to give base 49 a relative wealth of divisibility tests: only about half (51.02%) of all enneaquadragesimal digits are opaque. However, the only way to encode base 49 is through septenary, and with 49 being the square of 7, wouldnâ€™t we be using septenary itself? Digit Map
Intuitive Divisibility Tests
Regular Digits
Sorry, icarus; I couldnâ€™t resist. 

icarus 
Posted: Oct 4 2013, 02:58 AM

Dozens Demigod Group: Admin Posts: 1,913 Member No.: 50 Joined: 11April 06 
Holy cats oschkar that's awesome! (was one on my mind, also square of 11, 13, 17, 19. Latter two interesting). Now I got some stocked so lets see if we're on the same wavelength! Mine aren't so "random" though.

Stella of the Sapphire 
Posted: Oct 4 2013, 04:33 AM

Casual Member Group: Members Posts: 41 Member No.: 717 Joined: 22April 13 
Now here's my turn to post some more, in no particular order!
Base 56; 2*3*E (decimal, 66.) THE GOOD This is divisible by six, so halves and thirds are clean and clear! No greater number has 18; or fewer totatives. (Very small advantage) It's omega 55; is divisble by five THE BAD! Not divisible by four means that two places are needed for fourths, which is worse for a large base like this! Not divisible by five! Fifths do not have finite representation! THE UGLY!! Very weak ratio of divisors to totatives! 8:18; == 2:5 this is much worse than decimal (4:4==1:1) and means that too many figures require indirect manipulation, e.g. addition of two divisors, to work well. And abbreviated multiplication table simplifies use of the base at the heavy cost of more steps to solve a given problem!!! *RATING* 2/6 if 0~55; 2~3/6 if encoded six by eleven, 2~3/6 if balanced 29;~+29;, 3/6 balanced 3~+3 by 5~+5 Base 180; 2^4*3*5 (decimal, 240.) THE GOOD (Very strong advantage) Highly composite number! Very many divisors, important fractions are short and clear! Only 156; of smaller positive integers has a smaller totient ratio (8:2E;==20:89 vs 4:13==24:89, a ratio of 6:7) THE BAD! (Small disadvantage) Primeflank, so no short periods of recurrent fractional representations. THE UGLY!! Only problem I can think of is that it may be too big even for an abbreviated multiplication table by the RDM (reciprocal divisor method) *RATING* 4/6 if 0~17E; 4~5/6 if encoded twelve by twenty, 5/6 if balanced X0;~+X0;, 5/6 balanced 6~+6 by X~+X Base eleven THE GOOD Prime number base is good for padic representation and useful in cryptography (Small advantage) Eleven has alpha and omega divisors in {2,3,5} THE BAD! Not divisible by three! Not divisible by five! THE UGLY!! (Very strong disadvantage!) Odd number base, halves can not be exact without special notation!! *RATING* 0~1/6 in ALL representations except possibly 2~3/6 in niche situations. Definitely not a general use base Base 22; 2*11; (decimal, 26.) THE GOOD Unique positive integer with alpha *and* omega square or higher powers! Strong approximation to square root of three, 13:22:39; == (decimal)15:26:45 with very small error 484:483; == (decimal)676:675 THE BAD! Not divisible by three! Thirds recur, even with a short period of two for three squared and cubed! Not divisible by five! Fifths recur, even with short period of one for five squared! Fourths require two places past the fraction point! THE UGLY!! Requires special notation for recurrent figures to be useful, therefore not easy to comprehend for an ordinary person! *RATING* 2/6 standard, 3/6 balanced 11;~+11, 3~4/6 with special notation for recurrent figures. I would NOT use this unless special circumstances make it useful! And now for quadratic integers! hehehe Base \(1+\sqrt{2}\) THE GOOD Every integer in the ring Z(sqrt(2)) can be represented exactly! THE BAD! Nonunique representation of some numbers! THE UGLY!! No fraction can be represented exactly! Very unintuitive, try wrapping one's mind around this without getting a headache! *RATING* 1/6 with {0,1,2} where 100=21, 2/6 with {1,0,1} Base \(2+\sqrt{3}\) THE GOOD Every integer in the ring Z(sqrt(3)) can be represented exactly! THE BAD! Nonunique representation of some numbers! THE UGLY!! No fraction can be represented exactly! Very unintuitive, try wrapping one's mind around this without getting a headache! *RATING* 1/6 with {0,1,2,3} where 101=40, 1~2/6 with {2,1,0,1,2} Base \(\sqrt{3}\) THE GOOD Can represent square roots of negative numbers! Great way to boggle someone's mind! THE BAD! The set of figures can limit the utility of this base! THE UGLY!! Totally unwieldy for normal purposes! Requires heavy mental effort to make simple calculations with small numbers! *RATING* ???/6 I don't even know where to start! Â·Â·brain freezeÂ·Â· ...and yeah. You may call me crazy, but I don't care, because I just wanted to put in something here! 
wendy.krieger 
Posted: Oct 4 2013, 07:28 AM

Dozens Demigod Group: Members Posts: 2,432 Member No.: 655 Joined: 11July 12 
Base J(4) = \( 2+\sqrt{3} \) isn't all that bad. All of the integers have periods, and all periodic cycles divide some integer.
It's quite doable on the stone board, when one uses the digits 0, 1, 2, R, where R+1=10, and thus \(R=1+\sqrt{3} \). One notes that \(R^2=2.0 \). While it's not recommended as a general purpose base, it has its uses when grappling with duodecagonal isomorphisms. It has one known sevenite under 4147200000 (20*120^4), being 103. Likewise J(6) = \(1+\sqrt{2}\) is used for dealing with octagons etc. It's actually the square root of J(6), which means that periods are generally double what is indicated. This base has three sevenites to 20*120^4, being 13, 31, and some prime around 1.5*10^6. The first is very famous because it turns up in the 5:12:13 triangle. The digitset for this is \(0, 1, q = \sqrt{2} \). Of the three that geometrically occur this is the hardest one. Base 66 is unusual in that it has a regular of the second lowest form 1.0.18. 
icarus 
Posted: Oct 4 2013, 12:09 PM


Dozens Demigod Group: Admin Posts: 1,913 Member No.: 50 Joined: 11April 06 
Base 51  Unquinquagesimal (Dozenal: Base 43;  quadtrimal) The integer 51 is a semiprime {3, 17}, thus it has 4 divisor digits {0, 1, 3, 17}. Its totient ratio is (2/3 × 16/17) = 32/51 = 62.75% (76;43 p/g., 75:35 twelftieths). The unquinquagesimal digits that are not multiples of at least one of the prime divisors {3, 17} are coprime to 51. The ωnumber is 50 (dozenal 42) with factors {2, 5²}. The αnumber is the number 52 (dozenal 44) = {2², 13}. Thus conceptually, we can use the omega compound divisibility test on semicoprime numbers that are products of any positive power of the prime divisors {7, 17} and one of the powers 0 < τ ≤ 2 of t_{α} = {2, 5} and the other neighbor factor {13}. Base 51 is a rather typical semiprime with some fairly divisible neighbors. Digit Map
Intuitive Divisibility Rules
Regular Digits


anirudhk 
Posted: Oct 4 2013, 02:20 PM

Casual Member Group: Members Posts: 105 Member No.: 739 Joined: 8August 13 
Wow, that sure is a lot of bases!

anirudhk 
Posted: Oct 4 2013, 02:30 PM

Casual Member Group: Members Posts: 105 Member No.: 739 Joined: 8August 13 
Base 404 ( 298 in dozenal )
I just chose this base as it's similar to 12, as it is also 4 times a prime. But 12 is much more useful compared to this. Pros It's divisible by 4. Has a really awesome neighbor, providing divisibility tests for 3 up till it's 4th power and also 5. Cons Really hard to code (as 101 needs to be used as a subbase). Not divisible by 3. Not divisible by 5. Not even divisible by 7. Has only 6 factors, and few regular numbers. I would rate this base 1.9/6 stars. 
icarus 
Posted: Oct 4 2013, 09:54 PM


Dozens Demigod Group: Admin Posts: 1,913 Member No.: 50 Joined: 11April 06 
Base 289  Ducentenneoctagesimal (Dozenal: Base 201;) The integer 289 is the square of 17. Thus it has 3 divisor digits {0, 1, 17}. Its totient ratio is 16/17 = 28.57% (35.19 p/g., E3:E3 twelftieths). The ducentenneoctagesimal digits that are multiples of 17 are not coprime to 289. Base 289 has a sparse quantity of divisors and semitotatives swamped by 272 totatives. The ωnumber is the highly divisible 288 (dozenal 200) with its 18 factors {1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 72, 96, 144, 288}, products of primes {2, 3}. The αnumber is 290 (dozenal 202), with factors {1, 2, 5, 10, 29, 58, 145, 290}, products of primes {2, 5, 29}. Thus conceptually, we can use the omega compound divisibility test on semicoprime numbers that are products of any positive power of 17 and one of the powers 0 < τ ≤ 4 of t_{ω} = 2, and one of the powers 0 < τ ≤ 2 of t_{ω} = 3. We can use the alpha compound divisibility tests on semicoprime numbers that are products of any positive power of 17 and {2, 5, 29}. The neighbors of this large prime square convey a magnificent number of divisibility tests covering the primes {2, 3, 5, 17, 29}. Digit Map
Intuitive Divisibility Tests
Regular Digits


icarus 
Posted: Oct 4 2013, 10:17 PM


Dozens Demigod Group: Admin Posts: 1,913 Member No.: 50 Joined: 11April 06 
Base 169  Centenneasexagesimal (Dozenal: Base 121;) The integer 169 is the square of 13. Thus it has 3 divisor digits {0, 1, 13}. Its totient ratio is 12/13 = 92.31% (E1.E2 p/g., E0:92 twelftieths). The centenneasexagesimal digits that are multiples of 13 are not coprime to 169. Base 169 has a sparse quantity of divisors and semitotatives swamped by 156 totatives. The ωnumber is the highly divisible 168 (dozenal 120) with its factors {1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, 168}, products of primes {2, 3, 7}. The αnumber is 170 (dozenal 122), with factors {1, 2, 5, 10, 17, 34, 85, 170}, products of primes {2, 5, 17}. Thus conceptually, we can use the omega compound divisibility test on semicoprime numbers that are products of any positive power of 13 and one of the powers 0 < τ ≤ 2 of t_{ω} = 2, and {3, 7}. We can use the alpha compound divisibility tests on semicoprime numbers that are products of any positive power of 13 and {2, 5, 17}. The neighbors of this large prime square convey a magnificent number of divisibility tests covering the primes {2, 3, 5, 7, 13, 17}. Digit Map
Intuitive Divisibility Tests
Regular Digits


Oschkar 
Posted: Oct 4 2013, 11:03 PM

Dozens Disciple Group: Members Posts: 575 Member No.: 623 Joined: 19November 11 
Why do you have divisibility maps distinct from digit maps? As far as I can tell, the divisibility maps show all the information contained in the digit map, and much more...

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