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Pages: (4) 1 [2] 3 4  ( Go to first unread post )

 Random Bases
icarus
Posted: Oct 1 2013, 12:15 PM


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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 ω-number is 84 (dozenal 70) with a dozen factors {1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84}. The α-number 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

  _0 _1 _2 _3 _4 _5 _6 _7 _8 _9 _a _b _c _d _e _f _g
0_ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1_ 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
2_ 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
3_ 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67
4_ 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84

Intuitive Divisibility Tests

  _0 _1 _2 _3 _4 _5 _6 _7 _8 _9 _a _b _c _d _e _f _g
0_ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1_ 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
2_ 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
3_ 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67
4_ 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84

Regular Digits

  50 51 52
170 1 5 25
171 17 85  
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icarus
Posted: Oct 1 2013, 12:58 PM


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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 semi-random bases to study in that weird sequence.
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icarus
Posted: Oct 1 2013, 12:59 PM


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Base 27 | Heptavigesimal (Dozenal: Base 23; | bitrimal)

The integer 27 is the cube of three. As a prime power pε, 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 ω-number is 26 (dozenal 22) with factors {1, 2, 13, 26}. The α-number 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

  _0 _1 _2 _3 _4 _5 _6 _7 _8
0_ 0 1 2 3 4 5 6 7 8
1_ 9 10 11 12 13 14 15 16 17
2_ 18 19 20 21 22 23 24 25 26

Intuitive Divisibility Rules

  _0 _1 _2 _3 _4 _5 _6 _7 _8
0_ 0 1 2 3 4 5 6 7 8
1_ 9 10 11 12 13 14 15 16 17
2_ 18 19 20 21 22 23 24 25 26

Regular Digits

30 31 32 33
1 3 9 27
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icarus
Posted: Oct 2 2013, 11:44 AM


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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 ω-number is the highly divisible 132 (dozenal E0;). The α-number is the semiprime 134 (dozenal E2;) = {2, 67}. Thus conceptually, we can use compound divisibility tests on semi-coprime 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

  _0 _1 _2 _3 _4 _5 _6 _7 _8 _9 _a _b _c _d _e _f _g _h  
0_ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
1_ 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
2_ 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56
3_ 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75
4_ 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94
5_ 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113
6_ 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

Intuitive Divisibility Tests

  _0 _1 _2 _3 _4 _5 _6 _7 _8 _9 _a _b _c _d _e _f _g _h  
0_ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
1_ 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
2_ 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56
3_ 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75
4_ 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94
5_ 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113
6_ 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

Regular Digits

  70 71 72
190 1 7 49
191 19 133  
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dgoodmaniii
Posted: Oct 2 2013, 01:05 PM


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QUOTE (icarus @ Oct 2 2013, 11:44 AM)
(Dozenal: Base E1; | elunimal; “Base Lee”)

Levunimal, actually. But "Base Lee" still works.
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anirudhk
Posted: Oct 2 2013, 01:08 PM


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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.
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icarus
Posted: Oct 2 2013, 05:28 PM


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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 σ0(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 ω-number is the prime 503 (dozenal 35E). The α-number is the semiprime 505 (dozenal 361). Thus conceptually, we can use compound divisibility tests on semi-coprime 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

  0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
1 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47
2 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71
3 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95
4 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119
5 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143
6 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167
7 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191
8 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215
9 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239
10 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263
11 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287
12 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311
13 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335
14 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359
15 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383
16 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407
17 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431
18 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455
19 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479
20 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503
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icarus
Posted: Oct 2 2013, 05:32 PM


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[more 504]

Intuitive Divisibility Rules

  0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
1 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47
2 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71
3 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95
4 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119
5 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143
6 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167
7 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191
8 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215
9 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239
10 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263
11 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287
12 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311
13 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335
14 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359
15 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383
16 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407
17 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431
18 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455
19 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479
20 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503

Regular Digits

50 71 72 73
  20 21 22 23 24 25 26 27 28
30 1 2 4 8 16 32 64 128 256
31 3 6 12 24 48 96 192 384  
32 9 18 36 72 144 288      
33 27 54 108 216 432        
34 81 162 324            
35 243 486              
  20 21 22 23 24 25 26
30 7 14 28 56 112 224 448
31 21 42 84 168 336    
32 63 126 252 504      
33 189 378          
  20 21 22 23
30 49 98 196 392
31 147 294    
32 441      
  20
30 343
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wendy.krieger
Posted: Oct 3 2013, 07:24 AM


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Base 504 was one of the bases considered here. This was a program run to find the size of regulars against their 'co-regulars' (ie recriprocals). Base 504 has a very large plateau and sharp sides, due to 7, 8, 9 being pretty much the same size.

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icarus
Posted: Oct 3 2013, 11:29 AM


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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 ω-number is 28 (dozenal 24) with factors {1, 2, 4, 7, 14, 28}. The α-number 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

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

Intuitive Divisibility Rules

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

Regular Digits

290 291
1 29
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icarus
Posted: Oct 3 2013, 07:06 PM


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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 ω-number is 118 (dozenal 9X) with factors {2, 59}. The α-number is the colossally abundant number 120 (dozenal X0) = {2³, 3, 5}. Thus conceptually, we can use the omega compound divisibility test on semi-coprime numbers that are products of any positive power of the prime divisors {7, 17} and one of the powers 0 < τ ≤ 3 of tα = 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

  _0 _1 _2 _3 _4 _5 _6 _7 _8 _9 _a _b _c _d _e _f _g
0_ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1_ 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
2_ 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
3_ 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67
4_ 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84
5_ 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101
6_ 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118

Intuitive Divisibility Rules

  _0 _1 _2 _3 _4 _5 _6 _7 _8 _9 _a _b _c _d _e _f _g
0_ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1_ 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
2_ 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
3_ 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67
4_ 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84
5_ 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101
6_ 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118

Regular Digits

  70 71 72
170 1 7 49
171 17 119  
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dgoodmaniii
Posted: Oct 3 2013, 08:23 PM


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QUOTE (icarus @ Oct 3 2013, 07:06 PM)

Base 119 | Centononadecimal (Dozenal: Base 9E; | novlevimal)


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.
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icarus
Posted: Oct 3 2013, 09:07 PM


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wink.gif 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!
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icarus
Posted: Oct 3 2013, 09:12 PM


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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 ω-number is 39 (dozenal 33) with factors {1, 3, 13, 39}. The α-number 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

  _0 _1 _2 _3 _4 _5 _6 _7
0_ 0 1 2 3 4 5 6 7
1_ 8 9 10 11 12 13 14 15
2_ 16 17 18 19 20 21 22 23
3_ 24 25 26 27 28 29 30 31
4_ 32 33 34 35 36 37 38 39

Intuitive Divisibility Tests

  _0 _1 _2 _3 _4 _5 _6 _7
0_ 0 1 2 3 4 5 6 7
1_ 8 9 10 11 12 13 14 15
2_ 16 17 18 19 20 21 22 23
3_ 24 25 26 27 28 29 30 31
4_ 32 33 34 35 36 37 38 39

Regular Digits

  20 21 22 23 24 25
50 1 2 4 8 16 32
51 5 10 20 40    
52 25          
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Oschkar
Posted: Oct 4 2013, 01:45 AM


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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 ω-number is 48, a highly composite number with ten factors {1, 2, 3, 4, 6, 8, 12, 16, 24, 48}, and the α-number 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, wouldn’t we be using septenary itself?

Digit Map

  _0 _1 _2 _3 _4 _5 _6
0_ 0 1 2 3 4 5 6
1_ 7 8 9 10 11 12 13
2_ 14 15 16 17 18 19 20
3_ 21 22 23 24 25 26 27
4_ 28 29 30 31 32 33 34
5_ 35 36 37 38 39 40 41
6_ 42 43 44 45 46 47 48

Intuitive Divisibility Tests

  _0 _1 _2 _3 _4 _5 _6
0_ 0 1 2 3 4 5 6
1_ 7 8 9 10 11 12 13
2_ 14 15 16 17 18 19 20
3_ 21 22 23 24 25 26 27
4_ 28 29 30 31 32 33 34
5_ 35 36 37 38 39 40 41
6_ 42 43 44 45 46 47 48

Regular Digits

70 71 72
1 7 49

Sorry, icarus; I couldn’t resist.
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icarus
Posted: Oct 4 2013, 02:58 AM


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


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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 p-adic 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!
Non-unique 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!
Non-unique 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! ··brain freeze··


...and yeah. You may call me crazy, but I don't care, because I just wanted to put in something here!
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wendy.krieger
Posted: Oct 4 2013, 07:28 AM


Dozens Demigod


Group: Members
Posts: 2,432
Member No.: 655
Joined: 11-July 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 do-able 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 digit-set 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.



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icarus
Posted: Oct 4 2013, 12:09 PM


Dozens Demigod


Group: Admin
Posts: 1,913
Member No.: 50
Joined: 11-April 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 ω-number is 50 (dozenal 42) with factors {2, 5²}. The α-number is the number 52 (dozenal 44) = {2², 13}. Thus conceptually, we can use the omega compound divisibility test on semi-coprime numbers that are products of any positive power of the prime divisors {7, 17} and one of the powers 0 < τ ≤ 2 of tα = {2, 5} and the other neighbor factor {13}. Base 51 is a rather typical semiprime with some fairly divisible neighbors.

Digit Map

  _0 _1 _2 _3 _4 _5 _6 _7 _8 _9 _a _b _c _d _e _f _g
0_ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1_ 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
2_ 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

Intuitive Divisibility Rules

  _0 _1 _2 _3 _4 _5 _6 _7 _8 _9 _a _b _c _d _e _f _g
0_ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1_ 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
2_ 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

Regular Digits

  30 31 32 33
170 1 3 9 27
171 17 51    
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anirudhk
Posted: Oct 4 2013, 02:20 PM


Casual Member


Group: Members
Posts: 105
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Joined: 8-August 13



Wow, that sure is a lot of bases!
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anirudhk
Posted: Oct 4 2013, 02:30 PM


Casual Member


Group: Members
Posts: 105
Member No.: 739
Joined: 8-August 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 sub-base).
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.
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icarus
Posted: Oct 4 2013, 09:54 PM


Dozens Demigod


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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 ω-number 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 α-number 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 semi-coprime numbers that are products of any positive power of 17 and one of the powers 0 < τ ≤ 4 of tω = 2, and one of the powers 0 < τ ≤ 2 of tω = 3. We can use the alpha compound divisibility tests on semi-coprime 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

  _0 _1 _2 _3 _4 _5 _6 _7 _8 _9 _a _b _c _d _e _f _g
0_ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1_ 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
2_ 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
3_ 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67
4_ 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84
5_ 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101
6_ 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118
7_ 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135
8_ 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152
9_ 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169
a_ 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186
b_ 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203
c_ 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220
d_ 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237
e_ 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254
f_ 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271
g_ 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288

Intuitive Divisibility Tests

  _0 _1 _2 _3 _4 _5 _6 _7 _8 _9 _a _b _c _d _e _f _g
0_ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1_ 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
2_ 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
3_ 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67
4_ 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84
5_ 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101
6_ 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118
7_ 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135
8_ 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152
9_ 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169
a_ 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186
b_ 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203
c_ 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220
d_ 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237
e_ 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254
f_ 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271
g_ 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288

Regular Digits

170 171 172
1 17 289
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icarus
Posted: Oct 4 2013, 10:17 PM


Dozens Demigod


Group: Admin
Posts: 1,913
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Joined: 11-April 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 ω-number 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 α-number 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 semi-coprime numbers that are products of any positive power of 13 and one of the powers 0 < τ ≤ 2 of tω = 2, and {3, 7}. We can use the alpha compound divisibility tests on semi-coprime 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

  _0 _1 _2 _3 _4 _5 _6 _7 _8 _9 _a _b _c
0_ 0 1 2 3 4 5 6 7 8 9 10 11 12
1_ 13 14 15 16 17 18 19 20 21 22 23 24 25
2_ 26 27 28 29 30 31 32 33 34 35 36 37 38
3_ 39 40 41 42 43 44 45 46 47 48 49 50 51
4_ 52 53 54 55 56 57 58 59 60 61 62 63 64
5_ 65 66 67 68 69 70 71 72 73 74 75 76 77
6_ 78 79 80 81 82 83 84 85 86 87 88 89 90
7_ 91 92 93 94 95 96 97 98 99 100 101 102 103
8_ 104 105 106 107 108 109 110 111 112 113 114 115 116
9_ 117 118 119 120 121 122 123 124 125 126 127 128 129
a_ 130 131 132 133 134 135 136 137 138 139 140 141 142
b_ 143 144 145 146 147 148 149 150 151 152 153 154 155
c_ 156 157 158 159 160 161 162 163 164 165 166 167 168

Intuitive Divisibility Tests

  _0 _1 _2 _3 _4 _5 _6 _7 _8 _9 _a _b _c
0_ 0 1 2 3 4 5 6 7 8 9 10 11 12
1_ 13 14 15 16 17 18 19 20 21 22 23 24 25
2_ 26 27 28 29 30 31 32 33 34 35 36 37 38
3_ 39 40 41 42 43 44 45 46 47 48 49 50 51
4_ 52 53 54 55 56 57 58 59 60 61 62 63 64
5_ 65 66 67 68 69 70 71 72 73 74 75 76 77
6_ 78 79 80 81 82 83 84 85 86 87 88 89 90
7_ 91 92 93 94 95 96 97 98 99 100 101 102 103
8_ 104 105 106 107 108 109 110 111 112 113 114 115 116
9_ 117 118 119 120 121 122 123 124 125 126 127 128 129
a_ 130 131 132 133 134 135 136 137 138 139 140 141 142
b_ 143 144 145 146 147 148 149 150 151 152 153 154 155
c_ 156 157 158 159 160 161 162 163 164 165 166 167 168

Regular Digits

130 131 132
1 13 169
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Oschkar
Posted: Oct 4 2013, 11:03 PM


Dozens Disciple


Group: Members
Posts: 575
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Joined: 19-November 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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