Length 1120 sequences: Difference between revisions
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They have a lot of common structure. We can partition the numbers from 1 to 1120 in such a way that all sequences in Klas' list agree on each set in the partition, up to a change of sign. The largest of these sets has 729 members; the next largest 66. The sets in the partition, along with prime factorizations of their members, are listed below. | They have a lot of common structure. We can partition the numbers from 1 to 1120 in such a way that all sequences in Klas' list agree on each set in the partition, up to a change of sign. The largest of these sets has 729 members; the next largest 66. The sets in the partition, along with prime factorizations of their members, are listed below. | ||
The signs on the left represent one of the sequences. | |||
For example, all the sequences satisfy <math>-f(2) = +f(3) = +f(14) = -f(21) = +f(94) = -f(98) = -f(141) = +f(147)</math>. | |||
<pre> | <pre> | ||
************ | ************ | ||
1 | + 1 | ||
************ | ************ | ||
2 ( 2 ) | - 2 ( 2 ) | ||
3 ( 3 ) | + 3 ( 3 ) | ||
14 ( 2 ) ( 7 ) | + 14 ( 2 ) ( 7 ) | ||
21 ( 3 ) ( 7 ) | - 21 ( 3 ) ( 7 ) | ||
94 ( 2 ) ( 47 ) | + 94 ( 2 ) ( 47 ) | ||
98 ( 2 ) ( 7 ^ 2 ) | - 98 ( 2 ) ( 7 ^ 2 ) | ||
141 ( 3 ) ( 47 ) | - 141 ( 3 ) ( 47 ) | ||
147 ( 3 ) ( 7 ^ 2 ) | + 147 ( 3 ) ( 7 ^ 2 ) | ||
************ | ************ | ||
+ 4 ( 2 ^ 2 ) | |||
- 5 ( 5 ) | |||
- 6 ( 2 ) ( 3 ) | |||
- 8 ( 2 ^ 3 ) | |||
+ 9 ( 3 ^ 2 ) | |||
+ 10 ( 2 ) ( 5 ) | |||
+ 12 ( 2 ^ 2 ) ( 3 ) | |||
- 15 ( 3 ) ( 5 ) | |||
- 16 ( 2 ^ 4 ) | |||
- 18 ( 2 ) ( 3 ^ 2 ) | |||
+ 20 ( 2 ^ 2 ) ( 5 ) | |||
- 22 ( 2 ) ( 11 ) | |||
+ 24 ( 2 ^ 3 ) ( 3 ) | |||
- 25 ( 5 ^ 2 ) | |||
+ 26 ( 2 ) ( 13 ) | |||
+ 27 ( 3 ^ 3 ) | |||
- 28 ( 2 ^ 2 ) ( 7 ) | |||
- 30 ( 2 ) ( 3 ) ( 5 ) | |||
+ 32 ( 2 ^ 5 ) | |||
+ 33 ( 3 ) ( 11 ) | |||
- 34 ( 2 ) ( 17 ) | |||
+ 35 ( 5 ) ( 7 ) | |||
- 36 ( 2 ^ 2 ) ( 3 ^ 2 ) | |||
+ 38 ( 2 ) ( 19 ) | |||
- 39 ( 3 ) ( 13 ) | |||
- 40 ( 2 ^ 3 ) ( 5 ) | |||
+ 42 ( 2 ) ( 3 ) ( 7 ) | |||
- 43 ( 43 ) | |||
+ 44 ( 2 ^ 2 ) ( 11 ) | |||
+ 45 ( 3 ^ 2 ) ( 5 ) | |||
- 46 ( 2 ) ( 23 ) | |||
- 48 ( 2 ^ 4 ) ( 3 ) | |||
+ 50 ( 2 ) ( 5 ^ 2 ) | |||
+ 51 ( 3 ) ( 17 ) | |||
- 52 ( 2 ^ 2 ) ( 13 ) | |||
- 53 ( 53 ) | |||
+ 54 ( 2 ) ( 3 ^ 3 ) | |||
- 55 ( 5 ) ( 11 ) | |||
+ 56 ( 2 ^ 3 ) ( 7 ) | |||
- 57 ( 3 ) ( 19 ) | |||
- 58 ( 2 ) ( 29 ) | |||
+ 59 ( 59 ) | |||
+ 60 ( 2 ^ 2 ) ( 3 ) ( 5 ) | |||
+ 62 ( 2 ) ( 31 ) | |||
- 63 ( 3 ^ 2 ) ( 7 ) | |||
- 64 ( 2 ^ 6 ) | |||
+ 65 ( 5 ) ( 13 ) | |||
- 66 ( 2 ) ( 3 ) ( 11 ) | |||
+ 68 ( 2 ^ 2 ) ( 17 ) | |||
+ 69 ( 3 ) ( 23 ) | |||
- 70 ( 2 ) ( 5 ) ( 7 ) | |||
+ 72 ( 2 ^ 3 ) ( 3 ^ 2 ) | |||
- 75 ( 3 ) ( 5 ^ 2 ) | |||
- 76 ( 2 ^ 2 ) ( 19 ) | |||
+ 77 ( 7 ) ( 11 ) | |||
+ 78 ( 2 ) ( 3 ) ( 13 ) | |||
+ 80 ( 2 ^ 4 ) ( 5 ) | |||
- 81 ( 3 ^ 4 ) | |||
- 84 ( 2 ^ 2 ) ( 3 ) ( 7 ) | |||
- 85 ( 5 ) ( 17 ) | |||
+ 86 ( 2 ) ( 43 ) | |||
+ 87 ( 3 ) ( 29 ) | |||
- 88 ( 2 | |||
************ | ************ | ||
7 ( 7 ) | + 7 ( 7 ) | ||
************ | ************ | ||
11 ( 11 ) | - 11 ( 11 ) | ||
************ | ************ | ||
13 ( 13 ) | - 13 ( 13 ) | ||
************ | ************ | ||
17 ( 17 ) | - 17 ( 17 ) | ||
119 ( 7 ) ( 17 ) | + 119 ( 7 ) ( 17 ) | ||
************ | ************ | ||
19 ( 19 ) | + 19 ( 19 ) | ||
************ | ************ | ||
23 ( 23 ) | + 23 ( 23 ) | ||
************ | ************ | ||
29 ( 29 ) | - 29 ( 29 ) | ||
31 ( 31 ) | + 31 ( 31 ) | ||
************ | ************ | ||
37 ( 37 ) | - 37 ( 37 ) | ||
41 ( 41 ) | + 41 ( 41 ) | ||
************ | ************ | ||
47 ( 47 ) | + 47 ( 47 ) | ||
49 ( 7 ^ 2 ) | - 49 ( 7 ^ 2 ) | ||
319 ( 11 ) ( 29 ) | + 319 ( 11 ) ( 29 ) | ||
341 ( 11 ) ( 31 ) | - 341 ( 11 ) ( 31 ) | ||
************ | ************ | ||
61 ( 61 ) | - 61 ( 61 ) | ||
67 ( 67 ) | + 67 ( 67 ) | ||
************ | ************ | ||
71 ( 71 ) | - 71 ( 71 ) | ||
************ | ************ | ||
73 ( 73 ) | + 73 ( 73 ) | ||
************ | ************ | ||
74 ( 2 ) ( 37 ) | + 74 ( 2 ) ( 37 ) | ||
82 ( 2 ) ( 41 ) | - 82 ( 2 ) ( 41 ) | ||
************ | ************ | ||
79 ( 79 ) | - 79 ( 79 ) | ||
************ | ************ | ||
83 ( 83 ) | + 83 ( 83 ) | ||
************ | ************ | ||
89 ( 89 ) | + 89 ( 89 ) | ||
91 ( 7 ) ( 13 ) | - 91 ( 7 ) ( 13 ) | ||
************ | ************ | ||
101 ( 101 ) | - 101 ( 101 ) | ||
103 ( 103 ) | + 103 ( 103 ) | ||
************ | ************ | ||
107 ( 107 ) | + 107 ( 107 ) | ||
************ | ************ | ||
109 ( 109 ) | - 109 ( 109 ) | ||
************ | ************ | ||
111 ( 3 ) ( 37 ) | - 111 ( 3 ) ( 37 ) | ||
118 ( 2 ) ( 59 ) | + 118 ( 2 ) ( 59 ) | ||
122 ( 2 ) ( 61 ) | - 122 ( 2 ) ( 61 ) | ||
123 ( 3 ) ( 41 ) | + 123 ( 3 ) ( 41 ) | ||
************ | ************ | ||
113 ( 113 ) | + 113 ( 113 ) | ||
************ | ************ | ||
121 ( 11 ^ 2 ) | - 121 ( 11 ^ 2 ) | ||
************ | ************ | ||
133 ( 7 ) ( 19 ) | - 133 ( 7 ) ( 19 ) | ||
************ | ************ | ||
134 ( 2 ) ( 67 ) | + 134 ( 2 ) ( 67 ) | ||
************ | ************ | ||
137 ( 137 ) | - 137 ( 137 ) | ||
************ | ************ | ||
142 ( 2 ) ( 71 ) | - 142 ( 2 ) ( 71 ) | ||
************ | ************ | ||
143 ( 11 ) ( 13 ) | + 143 ( 11 ) ( 13 ) | ||
************ | ************ | ||
146 ( 2 ) ( 73 ) | + 146 ( 2 ) ( 73 ) | ||
************ | ************ | ||
148 ( 2 ^ 2 ) ( 37 ) | - 148 ( 2 ^ 2 ) ( 37 ) | ||
152 ( 2 ^ 3 ) ( 19 ) | + 152 ( 2 ^ 3 ) ( 19 ) | ||
164 ( 2 ^ 2 ) ( 41 ) | - 164 ( 2 ^ 2 ) ( 41 ) | ||
166 ( 2 ) ( 83 ) | + 166 ( 2 ) ( 83 ) | ||
172 ( 2 ^ 2 ) ( 43 ) | + 172 ( 2 ^ 2 ) ( 43 ) | ||
173 ( 173 ) | - 173 ( 173 ) | ||
184 ( 2 ^ 3 ) ( 23 ) | - 184 ( 2 ^ 3 ) ( 23 ) | ||
185 ( 5 ) ( 37 ) | + 185 ( 5 ) ( 37 ) | ||
188 ( 2 ^ 2 ) ( 47 ) | + 188 ( 2 ^ 2 ) ( 47 ) | ||
190 ( 2 ) ( 5 ) ( 19 ) | - 190 ( 2 ) ( 5 ) ( 19 ) | ||
205 ( 5 ) ( 41 ) | + 205 ( 5 ) ( 41 ) | ||
213 ( 3 ) ( 71 ) | + 213 ( 3 ) ( 71 ) | ||
215 ( 5 ) ( 43 ) | - 215 ( 5 ) ( 43 ) | ||
218 ( 2 ) ( 109 ) | - 218 ( 2 ) ( 109 ) | ||
222 ( 2 ) ( 3 ) ( 37 ) | + 222 ( 2 ) ( 3 ) ( 37 ) | ||
228 ( 2 ^ 2 ) ( 3 ) ( 19 ) | - 228 ( 2 ^ 2 ) ( 3 ) ( 19 ) | ||
230 ( 2 ) ( 5 ) ( 23 ) | + 230 ( 2 ) ( 5 ) ( 23 ) | ||
235 ( 5 ) ( 47 ) | - 235 ( 5 ) ( 47 ) | ||
242 ( 2 ) ( 11 ^ 2 ) | - 242 ( 2 ) ( 11 ^ 2 ) | ||
244 ( 2 ^ 2 ) ( 61 ) | + 244 ( 2 ^ 2 ) ( 61 ) | ||
246 ( 2 ) ( 3 ) ( 41 ) | + 246 ( 2 ) ( 3 ) ( 41 ) | ||
249 ( 3 ) ( 83 ) | - 249 ( 3 ) ( 83 ) | ||
258 ( 2 ) ( 3 ) ( 43 ) | - 258 ( 2 ) ( 3 ) ( 43 ) | ||
259 ( 7 ) ( 37 ) | + 259 ( 7 ) ( 37 ) | ||
276 ( 2 ^ 2 ) ( 3 ) ( 23 ) | + 276 ( 2 ^ 2 ) ( 3 ) ( 23 ) | ||
282 ( 2 ) ( 3 ) ( 47 ) | - 282 ( 2 ) ( 3 ) ( 47 ) | ||
285 ( 3 ) ( 5 ) ( 19 ) | + 285 ( 3 ) ( 5 ) ( 19 ) | ||
286 ( 2 ) ( 11 ) ( 13 ) | + 286 ( 2 ) ( 11 ) ( 13 ) | ||
287 ( 7 ) ( 41 ) | - 287 ( 7 ) ( 41 ) | ||
323 ( 17 ) ( 19 ) | - 323 ( 17 ) ( 19 ) | ||
331 ( 331 ) | + 331 ( 331 ) | ||
333 ( 3 ^ 2 ) ( 37 ) | - 333 ( 3 ^ 2 ) ( 37 ) | ||
338 ( 2 ) ( 13 ^ 2 ) | - 338 ( 2 ) ( 13 ^ 2 ) | ||
339 ( 3 ) ( 113 ) | + 339 ( 3 ) ( 113 ) | ||
342 ( 2 ) ( 3 ^ 2 ) ( 19 ) | + 342 ( 2 ) ( 3 ^ 2 ) ( 19 ) | ||
345 ( 3 ) ( 5 ) ( 23 ) | - 345 ( 3 ) ( 5 ) ( 23 ) | ||
347 ( 347 ) | + 347 ( 347 ) | ||
353 ( 353 ) | - 353 ( 353 ) | ||
355 ( 5 ) ( 71 ) | + 355 ( 5 ) ( 71 ) | ||
363 ( 3 ) ( 11 ^ 2 ) | + 363 ( 3 ) ( 11 ^ 2 ) | ||
366 ( 2 ) ( 3 ) ( 61 ) | - 366 ( 2 ) ( 3 ) ( 61 ) | ||
369 ( 3 ^ 2 ) ( 41 ) | - 369 ( 3 ^ 2 ) ( 41 ) | ||
387 ( 3 ^ 2 ) ( 43 ) | + 387 ( 3 ^ 2 ) ( 43 ) | ||
407 ( 11 ) ( 37 ) | - 407 ( 11 ) ( 37 ) | ||
414 ( 2 ) ( 3 ^ 2 ) ( 23 ) | - 414 ( 2 ) ( 3 ^ 2 ) ( 23 ) | ||
423 ( 3 ^ 2 ) ( 47 ) | + 423 ( 3 ^ 2 ) ( 47 ) | ||
429 ( 3 ) ( 11 ) ( 13 ) | - 429 ( 3 ) ( 11 ) ( 13 ) | ||
451 ( 11 ) ( 41 ) | + 451 ( 11 ) ( 41 ) | ||
481 ( 13 ) ( 37 ) | + 481 ( 13 ) ( 37 ) | ||
507 ( 3 ) ( 13 ^ 2 ) | + 507 ( 3 ) ( 13 ^ 2 ) | ||
513 ( 3 ^ 3 ) ( 19 ) | - 513 ( 3 ^ 3 ) ( 19 ) | ||
533 ( 13 ) ( 41 ) | - 533 ( 13 ) ( 41 ) | ||
535 ( 5 ) ( 107 ) | + 535 ( 5 ) ( 107 ) | ||
575 ( 5 ^ 2 ) ( 23 ) | - 575 ( 5 ^ 2 ) ( 23 ) | ||
577 ( 577 ) | + 577 ( 577 ) | ||
611 ( 13 ) ( 47 ) | - 611 ( 13 ) ( 47 ) | ||
621 ( 3 ^ 3 ) ( 23 ) | + 621 ( 3 ^ 3 ) ( 23 ) | ||
623 ( 7 ) ( 89 ) | - 623 ( 7 ) ( 89 ) | ||
625 ( 5 ^ 4 ) | + 625 ( 5 ^ 4 ) | ||
633 ( 3 ) ( 211 ) | + 633 ( 3 ) ( 211 ) | ||
635 ( 5 ) ( 127 ) | - 635 ( 5 ) ( 127 ) | ||
637 ( 7 ^ 2 ) ( 13 ) | + 637 ( 7 ^ 2 ) ( 13 ) | ||
639 ( 3 ^ 2 ) ( 71 ) | - 639 ( 3 ^ 2 ) ( 71 ) | ||
1017 ( 3 ^ 2 ) ( 113 ) | - 1017 ( 3 ^ 2 ) ( 113 ) | ||
1056 ( 2 ^ 5 ) ( 3 ) ( 11 ) | - 1056 ( 2 ^ 5 ) ( 3 ) ( 11 ) | ||
1072 ( 2 ^ 4 ) ( 67 ) | + 1072 ( 2 ^ 4 ) ( 67 ) | ||
************ | ************ | ||
149 ( 149 ) | - 149 ( 149 ) | ||
************ | ************ | ||
151 ( 151 ) | - 151 ( 151 ) | ||
203 ( 7 ) ( 29 ) | - 203 ( 7 ) ( 29 ) | ||
217 ( 7 ) ( 31 ) | + 217 ( 7 ) ( 31 ) | ||
219 ( 3 ) ( 73 ) | - 219 ( 3 ) ( 73 ) | ||
373 ( 373 ) | + 373 ( 373 ) | ||
613 ( 613 ) | + 613 ( 613 ) | ||
************ | ************ | ||
157 ( 157 ) | - 157 ( 157 ) | ||
************ | ************ | ||
161 ( 7 ) ( 23 ) | - 161 ( 7 ) ( 23 ) | ||
************ | ************ | ||
163 ( 163 ) | + 163 ( 163 ) | ||
************ | ************ | ||
177 ( 3 ) ( 59 ) | - 177 ( 3 ) ( 59 ) | ||
183 ( 3 ) ( 61 ) | + 183 ( 3 ) ( 61 ) | ||
************ | ************ | ||
179 ( 179 ) | - 179 ( 179 ) | ||
************ | ************ | ||
181 ( 181 ) | - 181 ( 181 ) | ||
************ | ************ | ||
191 ( 191 ) | + 191 ( 191 ) | ||
************ | ************ | ||
193 ( 193 ) | - 193 ( 193 ) | ||
************ | ************ | ||
194 ( 2 ) ( 97 ) | - 194 ( 2 ) ( 97 ) | ||
206 ( 2 ) ( 103 ) | + 206 ( 2 ) ( 103 ) | ||
************ | ************ | ||
197 ( 197 ) | + 197 ( 197 ) | ||
************ | ************ | ||
201 ( 3 ) ( 67 ) | - 201 ( 3 ) ( 67 ) | ||
367 ( 367 ) | + 367 ( 367 ) | ||
************ | ************ | ||
209 ( 11 ) ( 19 ) | - 209 ( 11 ) ( 19 ) | ||
************ | ************ | ||
211 ( 211 ) | - 211 ( 211 ) | ||
************ | ************ | ||
233 ( 233 ) | - 233 ( 233 ) | ||
************ | ************ | ||
239 ( 239 ) | + 239 ( 239 ) | ||
************ | ************ | ||
251 ( 251 ) | - 251 ( 251 ) | ||
************ | ************ | ||
253 ( 11 ) ( 23 ) | + 253 ( 11 ) ( 23 ) | ||
************ | ************ | ||
257 ( 257 ) | - 257 ( 257 ) | ||
************ | ************ | ||
262 ( 2 ) ( 131 ) | + 262 ( 2 ) ( 131 ) | ||
263 ( 263 ) | - 263 ( 263 ) | ||
************ | ************ | ||
268 ( 2 ^ 2 ) ( 67 ) | - 268 ( 2 ^ 2 ) ( 67 ) | ||
************ | ************ | ||
269 ( 269 ) | + 269 ( 269 ) | ||
************ | ************ | ||
271 ( 271 ) | + 271 ( 271 ) | ||
************ | ************ | ||
274 ( 2 ) ( 137 ) | + 274 ( 2 ) ( 137 ) | ||
************ | ************ | ||
277 ( 277 ) | - 277 ( 277 ) | ||
************ | ************ | ||
278 ( 2 ) ( 139 ) | - 278 ( 2 ) ( 139 ) | ||
************ | ************ | ||
281 ( 281 ) | - 281 ( 281 ) | ||
************ | ************ | ||
283 ( 283 ) | + 283 ( 283 ) | ||
************ | ************ | ||
284 ( 2 ^ 2 ) ( 71 ) | + 284 ( 2 ^ 2 ) ( 71 ) | ||
************ | ************ | ||
289 ( 17 ^ 2 ) | - 289 ( 17 ^ 2 ) | ||
************ | ************ | ||
291 ( 3 ) ( 97 ) | + 291 ( 3 ) ( 97 ) | ||
************ | ************ | ||
293 ( 293 ) | + 293 ( 293 ) | ||
************ | ************ | ||
295 ( 5 ) ( 59 ) | - 295 ( 5 ) ( 59 ) | ||
************ | ************ | ||
303 ( 3 ) ( 101 ) | - 303 ( 3 ) ( 101 ) | ||
************ | ************ | ||
305 ( 5 ) ( 61 ) | + 305 ( 5 ) ( 61 ) | ||
************ | ************ | ||
307 ( 307 ) | - 307 ( 307 ) | ||
************ | ************ | ||
309 ( 3 ) ( 103 ) | - 309 ( 3 ) ( 103 ) | ||
************ | ************ | ||
311 ( 311 ) | + 311 ( 311 ) | ||
************ | ************ | ||
329 ( 7 ) ( 47 ) | - 329 ( 7 ) ( 47 ) | ||
343 ( 7 ^ 3 ) | + 343 ( 7 ^ 3 ) | ||
************ | ************ | ||
358 ( 2 ) ( 179 ) | + 358 ( 2 ) ( 179 ) | ||
************ | ************ | ||
359 ( 359 ) | + 359 ( 359 ) | ||
************ | ************ | ||
361 ( 19 ^ 2 ) | - 361 ( 19 ^ 2 ) | ||
************ | ************ | ||
362 ( 2 ) ( 181 ) | - 362 ( 2 ) ( 181 ) | ||
************ | ************ | ||
377 ( 13 ) ( 29 ) | + 377 ( 13 ) ( 29 ) | ||
403 ( 13 ) ( 31 ) | - 403 ( 13 ) ( 31 ) | ||
************ | ************ | ||
379 ( 379 ) | - 379 ( 379 ) | ||
************ | ************ | ||
382 ( 2 ) ( 191 ) | + 382 ( 2 ) ( 191 ) | ||
386 ( 2 ) ( 193 ) | - 386 ( 2 ) ( 193 ) | ||
************ | ************ | ||
383 ( 383 ) | - 383 ( 383 ) | ||
************ | ************ | ||
389 ( 389 ) | - 389 ( 389 ) | ||
************ | ************ | ||
391 ( 17 ) ( 23 ) | + 391 ( 17 ) ( 23 ) | ||
************ | ************ | ||
393 ( 3 ) ( 131 ) | - 393 ( 3 ) ( 131 ) | ||
************ | ************ | ||
394 ( 2 ) ( 197 ) | + 394 ( 2 ) ( 197 ) | ||
************ | ************ | ||
397 ( 397 ) | + 397 ( 397 ) | ||
************ | ************ | ||
398 ( 2 ) ( 199 ) | - 398 ( 2 ) ( 199 ) | ||
************ | ************ | ||
401 ( 401 ) | + 401 ( 401 ) | ||
************ | ************ | ||
402 ( 2 ) ( 3 ) ( 67 ) | + 402 ( 2 ) ( 3 ) ( 67 ) | ||
************ | ************ | ||
409 ( 409 ) | - 409 ( 409 ) | ||
************ | ************ | ||
411 ( 3 ) ( 137 ) | - 411 ( 3 ) ( 137 ) | ||
************ | ************ | ||
413 ( 7 ) ( 59 ) | - 413 ( 7 ) ( 59 ) | ||
************ | ************ | ||
417 ( 3 ) ( 139 ) | + 417 ( 3 ) ( 139 ) | ||
************ | ************ | ||
419 ( 419 ) | + 419 ( 419 ) | ||
************ | ************ | ||
421 ( 421 ) | - 421 ( 421 ) | ||
************ | ************ | ||
422 ( 2 ) ( 211 ) | + 422 ( 2 ) ( 211 ) | ||
************ | ************ | ||
426 ( 2 ) ( 3 ) ( 71 ) | - 426 ( 2 ) ( 3 ) ( 71 ) | ||
************ | ************ | ||
427 ( 7 ) ( 61 ) | + 427 ( 7 ) ( 61 ) | ||
************ | ************ | ||
431 ( 431 ) | - 431 ( 431 ) | ||
************ | ************ | ||
433 ( 433 ) | + 433 ( 433 ) | ||
************ | ************ | ||
437 ( 19 ) ( 23 ) | + 437 ( 19 ) ( 23 ) | ||
************ | ************ | ||
439 ( 439 ) | - 439 ( 439 ) | ||
************ | ************ | ||
449 ( 449 ) | - 449 ( 449 ) | ||
************ | ************ | ||
457 ( 457 ) | + 457 ( 457 ) | ||
************ | ************ | ||
461 ( 461 ) | - 461 ( 461 ) | ||
************ | ************ | ||
463 ( 463 ) | + 463 ( 463 ) | ||
************ | ************ | ||
466 ( 2 ) ( 233 ) | - 466 ( 2 ) ( 233 ) | ||
478 ( 2 ) ( 239 ) | + 478 ( 2 ) ( 239 ) | ||
************ | ************ | ||
467 ( 467 ) | + 467 ( 467 ) | ||
************ | ************ | ||
469 ( 7 ) ( 67 ) | + 469 ( 7 ) ( 67 ) | ||
************ | ************ | ||
479 ( 479 ) | - 479 ( 479 ) | ||
************ | ************ | ||
487 ( 487 ) | - 487 ( 487 ) | ||
************ | ************ | ||
491 ( 491 ) | + 491 ( 491 ) | ||
************ | ************ | ||
493 ( 17 ) ( 29 ) | + 493 ( 17 ) ( 29 ) | ||
551 ( 19 ) ( 29 ) | - 551 ( 19 ) ( 29 ) | ||
569 ( 569 ) | - 569 ( 569 ) | ||
************ | ************ | ||
497 ( 7 ) ( 71 ) | - 497 ( 7 ) ( 71 ) | ||
553 ( 7 ) ( 79 ) | + 553 ( 7 ) ( 79 ) | ||
************ | ************ | ||
499 ( 499 ) | - 499 ( 499 ) | ||
************ | ************ | ||
503 ( 503 ) | + 503 ( 503 ) | ||
509 ( 509 ) | - 509 ( 509 ) | ||
************ | ************ | ||
527 ( 17 ) ( 31 ) | + 527 ( 17 ) ( 31 ) | ||
************ | ************ | ||
529 ( 23 ^ 2 ) | - 529 ( 23 ^ 2 ) | ||
************ | ************ | ||
531 ( 3 ^ 2 ) ( 59 ) | + 531 ( 3 ^ 2 ) ( 59 ) | ||
************ | ************ | ||
537 ( 3 ) ( 179 ) | - 537 ( 3 ) ( 179 ) | ||
************ | ************ | ||
538 ( 2 ) ( 269 ) | + 538 ( 2 ) ( 269 ) | ||
************ | ************ | ||
541 ( 541 ) | + 541 ( 541 ) | ||
************ | ************ | ||
542 ( 2 ) ( 271 ) | - 542 ( 2 ) ( 271 ) | ||
************ | ************ | ||
543 ( 3 ) ( 181 ) | + 543 ( 3 ) ( 181 ) | ||
************ | ************ | ||
547 ( 547 ) | + 547 ( 547 ) | ||
549 ( 3 ^ 2 ) ( 61 ) | - 549 ( 3 ^ 2 ) ( 61 ) | ||
************ | ************ | ||
554 ( 2 ) ( 277 ) | + 554 ( 2 ) ( 277 ) | ||
************ | ************ | ||
557 ( 557 ) | - 557 ( 557 ) | ||
************ | ************ | ||
563 ( 563 ) | - 563 ( 563 ) | ||
************ | ************ | ||
566 ( 2 ) ( 283 ) | - 566 ( 2 ) ( 283 ) | ||
************ | ************ | ||
571 ( 571 ) | - 571 ( 571 ) | ||
************ | ************ | ||
589 ( 19 ) ( 31 ) | + 589 ( 19 ) ( 31 ) | ||
591 ( 3 ) ( 197 ) | - 591 ( 3 ) ( 197 ) | ||
************ | ************ | ||
597 ( 3 ) ( 199 ) | - 597 ( 3 ) ( 199 ) | ||
1102 ( 2 ) ( 19 ) ( 29 ) | + 1102 ( 2 ) ( 19 ) ( 29 ) | ||
1103 ( 1103 ) | - 1103 ( 1103 ) | ||
************ | ************ | ||
599 ( 599 ) | - 599 ( 599 ) | ||
************ | ************ | ||
601 ( 601 ) | + 601 ( 601 ) | ||
************ | ************ | ||
603 ( 3 ^ 2 ) ( 67 ) | + 603 ( 3 ^ 2 ) ( 67 ) | ||
************ | ************ | ||
607 ( 607 ) | + 607 ( 607 ) | ||
************ | ************ | ||
614 ( 2 ) ( 307 ) | - 614 ( 2 ) ( 307 ) | ||
622 ( 2 ) ( 311 ) | + 622 ( 2 ) ( 311 ) | ||
************ | ************ | ||
617 ( 617 ) | - 617 ( 617 ) | ||
************ | ************ | ||
619 ( 619 ) | + 619 ( 619 ) | ||
************ | ************ | ||
629 ( 17 ) ( 37 ) | + 629 ( 17 ) ( 37 ) | ||
631 ( 631 ) | - 631 ( 631 ) | ||
697 ( 17 ) ( 41 ) | - 697 ( 17 ) ( 41 ) | ||
************ | ************ | ||
641 ( 641 ) | + 641 ( 641 ) | ||
643 ( 643 ) | - 643 ( 643 ) | ||
************ | ************ | ||
647 ( 647 ) | - 647 ( 647 ) | ||
************ | ************ | ||
653 ( 653 ) | - 653 ( 653 ) | ||
************ | ************ | ||
659 ( 659 ) | - 659 ( 659 ) | ||
************ | ************ | ||
661 ( 661 ) | + 661 ( 661 ) | ||
************ | ************ | ||
683 ( 683 ) | - 683 ( 683 ) | ||
685 ( 5 ) ( 137 ) | + 685 ( 5 ) ( 137 ) | ||
695 ( 5 ) ( 139 ) | - 695 ( 5 ) ( 139 ) | ||
************ | ************ | ||
694 ( 2 ) ( 347 ) | + 694 ( 2 ) ( 347 ) | ||
698 ( 2 ) ( 349 ) | - 698 ( 2 ) ( 349 ) | ||
************ | ************ | ||
699 ( 3 ) ( 233 ) | + 699 ( 3 ) ( 233 ) | ||
717 ( 3 ) ( 239 ) | - 717 ( 3 ) ( 239 ) | ||
719 ( 719 ) | + 719 ( 719 ) | ||
************ | ************ | ||
701 ( 701 ) | - 701 ( 701 ) | ||
************ | ************ | ||
703 ( 19 ) ( 37 ) | + 703 ( 19 ) ( 37 ) | ||
773 ( 773 ) | + 773 ( 773 ) | ||
779 ( 19 ) ( 41 ) | - 779 ( 19 ) ( 41 ) | ||
************ | ************ | ||
709 ( 709 ) | + 709 ( 709 ) | ||
************ | ************ | ||
739 ( 739 ) | - 739 ( 739 ) | ||
743 ( 743 ) | + 743 ( 743 ) | ||
************ | ************ | ||
749 ( 7 ) ( 107 ) | - 749 ( 7 ) ( 107 ) | ||
751 ( 751 ) | + 751 ( 751 ) | ||
************ | ************ | ||
757 ( 757 ) | + 757 ( 757 ) | ||
761 ( 761 ) | - 761 ( 761 ) | ||
************ | ************ | ||
763 ( 7 ) ( 109 ) | - 763 ( 7 ) ( 109 ) | ||
769 ( 769 ) | + 769 ( 769 ) | ||
907 ( 907 ) | + 907 ( 907 ) | ||
************ | ************ | ||
788 ( 2 ^ 2 ) ( 197 ) | - 788 ( 2 ^ 2 ) ( 197 ) | ||
796 ( 2 ^ 2 ) ( 199 ) | + 796 ( 2 ^ 2 ) ( 199 ) | ||
************ | ************ | ||
793 ( 13 ) ( 61 ) | + 793 ( 13 ) ( 61 ) | ||
797 ( 797 ) | - 797 ( 797 ) | ||
863 ( 863 ) | + 863 ( 863 ) | ||
871 ( 13 ) ( 67 ) | - 871 ( 13 ) ( 67 ) | ||
************ | ************ | ||
807 ( 3 ) ( 269 ) | - 807 ( 3 ) ( 269 ) | ||
************ | ************ | ||
808 ( 2 ^ 3 ) ( 101 ) | + 808 ( 2 ^ 3 ) ( 101 ) | ||
818 ( 2 ) ( 409 ) | - 818 ( 2 ) ( 409 ) | ||
824 ( 2 ^ 3 ) ( 103 ) | - 824 ( 2 ^ 3 ) ( 103 ) | ||
826 ( 2 ) ( 7 ) ( 59 ) | + 826 ( 2 ) ( 7 ) ( 59 ) | ||
938 ( 2 ) ( 7 ) ( 67 ) | - 938 ( 2 ) ( 7 ) ( 67 ) | ||
************ | ************ | ||
809 ( 809 ) | + 809 ( 809 ) | ||
************ | ************ | ||
811 ( 811 ) | - 811 ( 811 ) | ||
************ | ************ | ||
813 ( 3 ) ( 271 ) | - 813 ( 3 ) ( 271 ) | ||
************ | ************ | ||
821 ( 821 ) | - 821 ( 821 ) | ||
************ | ************ | ||
823 ( 823 ) | + 823 ( 823 ) | ||
************ | ************ | ||
827 ( 827 ) | - 827 ( 827 ) | ||
************ | ************ | ||
829 ( 829 ) | + 829 ( 829 ) | ||
************ | ************ | ||
831 ( 3 ) ( 277 ) | + 831 ( 3 ) ( 277 ) | ||
************ | ************ | ||
857 ( 857 ) | + 857 ( 857 ) | ||
859 ( 859 ) | - 859 ( 859 ) | ||
************ | ************ | ||
862 ( 2 ) ( 431 ) | + 862 ( 2 ) ( 431 ) | ||
866 ( 2 ) ( 433 ) | - 866 ( 2 ) ( 433 ) | ||
************ | ************ | ||
883 ( 883 ) | + 883 ( 883 ) | ||
887 ( 887 ) | - 887 ( 887 ) | ||
************ | ************ | ||
901 ( 17 ) ( 53 ) | + 901 ( 17 ) ( 53 ) | ||
922 ( 2 ) ( 461 ) | + 922 ( 2 ) ( 461 ) | ||
926 ( 2 ) ( 463 ) | - 926 ( 2 ) ( 463 ) | ||
************ | ************ | ||
909 ( 3 ^ 2 ) ( 101 ) | - 909 ( 3 ^ 2 ) ( 101 ) | ||
927 ( 3 ^ 2 ) ( 103 ) | + 927 ( 3 ^ 2 ) ( 103 ) | ||
************ | ************ | ||
919 ( 919 ) | - 919 ( 919 ) | ||
************ | ************ | ||
921 ( 3 ) ( 307 ) | + 921 ( 3 ) ( 307 ) | ||
933 ( 3 ) ( 311 ) | - 933 ( 3 ) ( 311 ) | ||
************ | ************ | ||
923 ( 13 ) ( 71 ) | + 923 ( 13 ) ( 71 ) | ||
************ | ************ | ||
925 ( 5 ^ 2 ) ( 37 ) | - 925 ( 5 ^ 2 ) ( 37 ) | ||
************ | ************ | ||
932 ( 2 ^ 2 ) ( 233 ) | - 932 ( 2 ^ 2 ) ( 233 ) | ||
************ | ************ | ||
934 ( 2 ) ( 467 ) | + 934 ( 2 ) ( 467 ) | ||
************ | ************ | ||
937 ( 937 ) | + 937 ( 937 ) | ||
************ | ************ | ||
941 ( 941 ) | + 941 ( 941 ) | ||
************ | ************ | ||
947 ( 947 ) | + 947 ( 947 ) | ||
949 ( 13 ) ( 73 ) | - 949 ( 13 ) ( 73 ) | ||
************ | ************ | ||
953 ( 953 ) | - 953 ( 953 ) | ||
************ | ************ | ||
955 ( 5 ) ( 191 ) | + 955 ( 5 ) ( 191 ) | ||
************ | ************ | ||
956 ( 2 ^ 2 ) ( 239 ) | - 956 ( 2 ^ 2 ) ( 239 ) | ||
************ | ************ | ||
958 ( 2 ) ( 479 ) | - 958 ( 2 ) ( 479 ) | ||
************ | ************ | ||
959 ( 7 ) ( 137 ) | + 959 ( 7 ) ( 137 ) | ||
************ | ************ | ||
961 ( 31 ^ 2 ) | - 961 ( 31 ^ 2 ) | ||
************ | ************ | ||
962 ( 2 ) ( 13 ) ( 37 ) | + 962 ( 2 ) ( 13 ) ( 37 ) | ||
************ | ************ | ||
964 ( 2 ^ 2 ) ( 241 ) | + 964 ( 2 ^ 2 ) ( 241 ) | ||
************ | ************ | ||
965 ( 5 ) ( 193 ) | + 965 ( 5 ) ( 193 ) | ||
************ | ************ | ||
967 ( 967 ) | - 967 ( 967 ) | ||
************ | ************ | ||
971 ( 971 ) | - 971 ( 971 ) | ||
************ | ************ | ||
973 ( 7 ) ( 139 ) | - 973 ( 7 ) ( 139 ) | ||
************ | ************ | ||
977 ( 977 ) | + 977 ( 977 ) | ||
************ | ************ | ||
982 ( 2 ) ( 491 ) | + 982 ( 2 ) ( 491 ) | ||
************ | ************ | ||
983 ( 983 ) | + 983 ( 983 ) | ||
************ | ************ | ||
985 ( 5 ) ( 197 ) | - 985 ( 5 ) ( 197 ) | ||
995 ( 5 ) ( 199 ) | + 995 ( 5 ) ( 199 ) | ||
************ | ************ | ||
997 ( 997 ) | + 997 ( 997 ) | ||
************ | ************ | ||
998 ( 2 ) ( 499 ) | - 998 ( 2 ) ( 499 ) | ||
************ | ************ | ||
1003 ( 17 ) ( 59 ) | - 1003 ( 17 ) ( 59 ) | ||
************ | ************ | ||
1009 ( 1009 ) | - 1009 ( 1009 ) | ||
************ | ************ | ||
1013 ( 1013 ) | + 1013 ( 1013 ) | ||
************ | ************ | ||
1019 ( 1019 ) | - 1019 ( 1019 ) | ||
1021 ( 1021 ) | + 1021 ( 1021 ) | ||
************ | ************ | ||
1023 ( 3 ) ( 11 ) ( 31 ) | + 1023 ( 3 ) ( 11 ) ( 31 ) | ||
************ | ************ | ||
1027 ( 13 ) ( 79 ) | - 1027 ( 13 ) ( 79 ) | ||
************ | ************ | ||
1031 ( 1031 ) | + 1031 ( 1031 ) | ||
************ | ************ | ||
1033 ( 1033 ) | + 1033 ( 1033 ) | ||
************ | ************ | ||
1037 ( 17 ) ( 61 ) | - 1037 ( 17 ) ( 61 ) | ||
************ | ************ | ||
1039 ( 1039 ) | - 1039 ( 1039 ) | ||
************ | ************ | ||
1041 ( 3 ) ( 347 ) | - 1041 ( 3 ) ( 347 ) | ||
************ | ************ | ||
1044 ( 2 ^ 2 ) ( 3 ^ 2 ) ( 29 ) | + 1044 ( 2 ^ 2 ) ( 3 ^ 2 ) ( 29 ) | ||
************ | ************ | ||
1046 ( 2 ) ( 523 ) | - 1046 ( 2 ) ( 523 ) | ||
************ | ************ | ||
1047 ( 3 ) ( 349 ) | - 1047 ( 3 ) ( 349 ) | ||
************ | ************ | ||
1049 ( 1049 ) | - 1049 ( 1049 ) | ||
************ | ************ | ||
1051 ( 1051 ) | + 1051 ( 1051 ) | ||
************ | ************ | ||
1054 ( 2 ) ( 17 ) ( 31 ) | + 1054 ( 2 ) ( 17 ) ( 31 ) | ||
************ | ************ | ||
1055 ( 5 ) ( 211 ) | - 1055 ( 5 ) ( 211 ) | ||
************ | ************ | ||
1058 ( 2 ) ( 23 ^ 2 ) | + 1058 ( 2 ) ( 23 ^ 2 ) | ||
************ | ************ | ||
1059 ( 3 ) ( 353 ) | + 1059 ( 3 ) ( 353 ) | ||
************ | ************ | ||
1061 ( 1061 ) | - 1061 ( 1061 ) | ||
************ | ************ | ||
1062 ( 2 ) ( 3 ^ 2 ) ( 59 ) | - 1062 ( 2 ) ( 3 ^ 2 ) ( 59 ) | ||
1068 ( 2 ^ 2 ) ( 3 ) ( 89 ) | - 1068 ( 2 ^ 2 ) ( 3 ) ( 89 ) | ||
************ | ************ | ||
1063 ( 1063 ) | + 1063 ( 1063 ) | ||
************ | ************ | ||
1065 ( 3 ) ( 5 ) ( 71 ) | + 1065 ( 3 ) ( 5 ) ( 71 ) | ||
************ | ************ | ||
1066 ( 2 ) ( 13 ) ( 41 ) | + 1066 ( 2 ) ( 13 ) ( 41 ) | ||
************ | ************ | ||
1067 ( 11 ) ( 97 ) | - 1067 ( 11 ) ( 97 ) | ||
************ | ************ | ||
1069 ( 1069 ) | + 1069 ( 1069 ) | ||
************ | ************ | ||
1070 ( 2 ) ( 5 ) ( 107 ) | - 1070 ( 2 ) ( 5 ) ( 107 ) | ||
************ | ************ | ||
1073 ( 29 ) ( 37 ) | - 1073 ( 29 ) ( 37 ) | ||
************ | ************ | ||
1074 ( 2 ) ( 3 ) ( 179 ) | + 1074 ( 2 ) ( 3 ) ( 179 ) | ||
************ | ************ | ||
1076 ( 2 ^ 2 ) ( 269 ) | - 1076 ( 2 ^ 2 ) ( 269 ) | ||
************ | ************ | ||
1077 ( 3 ) ( 359 ) | - 1077 ( 3 ) ( 359 ) | ||
************ | ************ | ||
1079 ( 13 ) ( 83 ) | - 1079 ( 13 ) ( 83 ) | ||
************ | ************ | ||
1081 ( 23 ) ( 47 ) | - 1081 ( 23 ) ( 47 ) | ||
************ | ************ | ||
1082 ( 2 ) ( 541 ) | - 1082 ( 2 ) ( 541 ) | ||
************ | ************ | ||
1084 ( 2 ^ 2 ) ( 271 ) | + 1084 ( 2 ^ 2 ) ( 271 ) | ||
************ | ************ | ||
1087 ( 1087 ) | + 1087 ( 1087 ) | ||
************ | ************ | ||
1090 ( 2 ) ( 5 ) ( 109 ) | + 1090 ( 2 ) ( 5 ) ( 109 ) | ||
************ | ************ | ||
1091 ( 1091 ) | - 1091 ( 1091 ) | ||
************ | ************ | ||
1093 ( 1093 ) | + 1093 ( 1093 ) | ||
************ | ************ | ||
1094 ( 2 ) ( 547 ) | - 1094 ( 2 ) ( 547 ) | ||
************ | ************ | ||
1096 ( 2 ^ 3 ) ( 137 ) | - 1096 ( 2 ^ 3 ) ( 137 ) | ||
1112 ( 2 ^ 3 ) ( 139 ) | + 1112 ( 2 ^ 3 ) ( 139 ) | ||
************ | ************ | ||
1097 ( 1097 ) | + 1097 ( 1097 ) | ||
************ | ************ | ||
1101 ( 3 ) ( 367 ) | + 1101 ( 3 ) ( 367 ) | ||
************ | ************ | ||
1108 ( 2 ^ 2 ) ( 277 ) | + 1108 ( 2 ^ 2 ) ( 277 ) | ||
************ | ************ | ||
1109 ( 1109 ) | - 1109 ( 1109 ) | ||
************ | ************ | ||
1110 ( 2 ) ( 3 ) ( 5 ) ( 37 ) | - 1110 ( 2 ) ( 3 ) ( 5 ) ( 37 ) | ||
************ | ************ | ||
1111 ( 11 ) ( 101 ) | + 1111 ( 11 ) ( 101 ) | ||
************ | ************ | ||
1114 ( 2 ) ( 557 ) | + 1114 ( 2 ) ( 557 ) | ||
************ | ************ | ||
1117 ( 1117 ) | + 1117 ( 1117 ) | ||
************ | ************ | ||
1118 ( 2 ) ( 13 ) ( 43 ) | - 1118 ( 2 ) ( 13 ) ( 43 ) | ||
************ | ************ | ||
1119 ( 3 ) ( 373 ) | + 1119 ( 3 ) ( 373 ) | ||
</pre> | </pre> | ||
Latest revision as of 11:47, 8 October 2010
Klas produced a list of 64 length-1120 discrepancy-2 sequences generated by his program:
http://abel.math.umu.se/~klasm/Data/EDP/Sequences-C=2-n=1120/
They have a lot of common structure. We can partition the numbers from 1 to 1120 in such a way that all sequences in Klas' list agree on each set in the partition, up to a change of sign. The largest of these sets has 729 members; the next largest 66. The sets in the partition, along with prime factorizations of their members, are listed below.
The signs on the left represent one of the sequences.
For example, all the sequences satisfy [math]\displaystyle{ -f(2) = +f(3) = +f(14) = -f(21) = +f(94) = -f(98) = -f(141) = +f(147) }[/math].
************ + 1 ************ - 2 ( 2 ) + 3 ( 3 ) + 14 ( 2 ) ( 7 ) - 21 ( 3 ) ( 7 ) + 94 ( 2 ) ( 47 ) - 98 ( 2 ) ( 7 ^ 2 ) - 141 ( 3 ) ( 47 ) + 147 ( 3 ) ( 7 ^ 2 ) ************ + 4 ( 2 ^ 2 ) - 5 ( 5 ) - 6 ( 2 ) ( 3 ) - 8 ( 2 ^ 3 ) + 9 ( 3 ^ 2 ) + 10 ( 2 ) ( 5 ) + 12 ( 2 ^ 2 ) ( 3 ) - 15 ( 3 ) ( 5 ) - 16 ( 2 ^ 4 ) - 18 ( 2 ) ( 3 ^ 2 ) + 20 ( 2 ^ 2 ) ( 5 ) - 22 ( 2 ) ( 11 ) + 24 ( 2 ^ 3 ) ( 3 ) - 25 ( 5 ^ 2 ) + 26 ( 2 ) ( 13 ) + 27 ( 3 ^ 3 ) - 28 ( 2 ^ 2 ) ( 7 ) - 30 ( 2 ) ( 3 ) ( 5 ) + 32 ( 2 ^ 5 ) + 33 ( 3 ) ( 11 ) - 34 ( 2 ) ( 17 ) + 35 ( 5 ) ( 7 ) - 36 ( 2 ^ 2 ) ( 3 ^ 2 ) + 38 ( 2 ) ( 19 ) - 39 ( 3 ) ( 13 ) - 40 ( 2 ^ 3 ) ( 5 ) + 42 ( 2 ) ( 3 ) ( 7 ) - 43 ( 43 ) + 44 ( 2 ^ 2 ) ( 11 ) + 45 ( 3 ^ 2 ) ( 5 ) - 46 ( 2 ) ( 23 ) - 48 ( 2 ^ 4 ) ( 3 ) + 50 ( 2 ) ( 5 ^ 2 ) + 51 ( 3 ) ( 17 ) - 52 ( 2 ^ 2 ) ( 13 ) - 53 ( 53 ) + 54 ( 2 ) ( 3 ^ 3 ) - 55 ( 5 ) ( 11 ) + 56 ( 2 ^ 3 ) ( 7 ) - 57 ( 3 ) ( 19 ) - 58 ( 2 ) ( 29 ) + 59 ( 59 ) + 60 ( 2 ^ 2 ) ( 3 ) ( 5 ) + 62 ( 2 ) ( 31 ) - 63 ( 3 ^ 2 ) ( 7 ) - 64 ( 2 ^ 6 ) + 65 ( 5 ) ( 13 ) - 66 ( 2 ) ( 3 ) ( 11 ) + 68 ( 2 ^ 2 ) ( 17 ) + 69 ( 3 ) ( 23 ) - 70 ( 2 ) ( 5 ) ( 7 ) + 72 ( 2 ^ 3 ) ( 3 ^ 2 ) - 75 ( 3 ) ( 5 ^ 2 ) - 76 ( 2 ^ 2 ) ( 19 ) + 77 ( 7 ) ( 11 ) + 78 ( 2 ) ( 3 ) ( 13 ) + 80 ( 2 ^ 4 ) ( 5 ) - 81 ( 3 ^ 4 ) - 84 ( 2 ^ 2 ) ( 3 ) ( 7 ) - 85 ( 5 ) ( 17 ) + 86 ( 2 ) ( 43 ) + 87 ( 3 ) ( 29 ) - 88 ( 2 ^ 3 ) ( 11 ) - 90 ( 2 ) ( 3 ^ 2 ) ( 5 ) + 92 ( 2 ^ 2 ) ( 23 ) - 93 ( 3 ) ( 31 ) + 95 ( 5 ) ( 19 ) + 96 ( 2 ^ 5 ) ( 3 ) - 97 ( 97 ) + 99 ( 3 ^ 2 ) ( 11 ) - 100 ( 2 ^ 2 ) ( 5 ^ 2 ) - 102 ( 2 ) ( 3 ) ( 17 ) + 104 ( 2 ^ 3 ) ( 13 ) + 105 ( 3 ) ( 5 ) ( 7 ) - 106 ( 2 ) ( 53 ) - 108 ( 2 ^ 2 ) ( 3 ^ 3 ) + 110 ( 2 ) ( 5 ) ( 11 ) + 112 ( 2 ^ 4 ) ( 7 ) + 114 ( 2 ) ( 3 ) ( 19 ) - 115 ( 5 ) ( 23 ) - 116 ( 2 ^ 2 ) ( 29 ) - 117 ( 3 ^ 2 ) ( 13 ) - 120 ( 2 ^ 3 ) ( 3 ) ( 5 ) + 124 ( 2 ^ 2 ) ( 31 ) + 125 ( 5 ^ 3 ) + 126 ( 2 ) ( 3 ^ 2 ) ( 7 ) - 127 ( 127 ) - 128 ( 2 ^ 7 ) - 129 ( 3 ) ( 43 ) - 130 ( 2 ) ( 5 ) ( 13 ) + 131 ( 131 ) + 132 ( 2 ^ 2 ) ( 3 ) ( 11 ) + 135 ( 3 ^ 3 ) ( 5 ) + 136 ( 2 ^ 3 ) ( 17 ) - 138 ( 2 ) ( 3 ) ( 23 ) + 139 ( 139 ) - 140 ( 2 ^ 2 ) ( 5 ) ( 7 ) - 144 ( 2 ^ 4 ) ( 3 ^ 2 ) + 145 ( 5 ) ( 29 ) + 150 ( 2 ) ( 3 ) ( 5 ^ 2 ) + 153 ( 3 ^ 2 ) ( 17 ) + 154 ( 2 ) ( 7 ) ( 11 ) - 155 ( 5 ) ( 31 ) - 156 ( 2 ^ 2 ) ( 3 ) ( 13 ) - 158 ( 2 ) ( 79 ) + 159 ( 3 ) ( 53 ) + 160 ( 2 ^ 5 ) ( 5 ) + 162 ( 2 ) ( 3 ^ 4 ) - 165 ( 3 ) ( 5 ) ( 11 ) + 167 ( 167 ) - 168 ( 2 ^ 3 ) ( 3 ) ( 7 ) + 169 ( 13 ^ 2 ) - 170 ( 2 ) ( 5 ) ( 17 ) - 171 ( 3 ^ 2 ) ( 19 ) + 174 ( 2 ) ( 3 ) ( 29 ) + 175 ( 5 ^ 2 ) ( 7 ) - 176 ( 2 ^ 4 ) ( 11 ) + 178 ( 2 ) ( 89 ) + 180 ( 2 ^ 2 ) ( 3 ^ 2 ) ( 5 ) - 182 ( 2 ) ( 7 ) ( 13 ) - 186 ( 2 ) ( 3 ) ( 31 ) + 187 ( 11 ) ( 17 ) - 189 ( 3 ^ 3 ) ( 7 ) + 192 ( 2 ^ 6 ) ( 3 ) + 195 ( 3 ) ( 5 ) ( 13 ) + 196 ( 2 ^ 2 ) ( 7 ^ 2 ) - 198 ( 2 ) ( 3 ^ 2 ) ( 11 ) + 199 ( 199 ) - 200 ( 2 ^ 3 ) ( 5 ^ 2 ) + 202 ( 2 ) ( 101 ) - 204 ( 2 ^ 2 ) ( 3 ) ( 17 ) + 207 ( 3 ^ 2 ) ( 23 ) + 208 ( 2 ^ 4 ) ( 13 ) + 210 ( 2 ) ( 3 ) ( 5 ) ( 7 ) - 212 ( 2 ^ 2 ) ( 53 ) - 214 ( 2 ) ( 107 ) + 216 ( 2 ^ 3 ) ( 3 ^ 3 ) + 220 ( 2 ^ 2 ) ( 5 ) ( 11 ) - 221 ( 13 ) ( 17 ) + 223 ( 223 ) - 224 ( 2 ^ 5 ) ( 7 ) - 225 ( 3 ^ 2 ) ( 5 ^ 2 ) - 226 ( 2 ) ( 113 ) + 227 ( 227 ) + 229 ( 229 ) - 231 ( 3 ) ( 7 ) ( 11 ) + 232 ( 2 ^ 3 ) ( 29 ) + 234 ( 2 ) ( 3 ^ 2 ) ( 13 ) - 236 ( 2 ^ 2 ) ( 59 ) + 237 ( 3 ) ( 79 ) + 238 ( 2 ) ( 7 ) ( 17 ) - 240 ( 2 ^ 4 ) ( 3 ) ( 5 ) + 241 ( 241 ) - 243 ( 3 ^ 5 ) - 245 ( 5 ) ( 7 ^ 2 ) + 247 ( 13 ) ( 19 ) - 248 ( 2 ^ 3 ) ( 31 ) + 250 ( 2 ) ( 5 ^ 3 ) + 252 ( 2 ^ 2 ) ( 3 ^ 2 ) ( 7 ) - 254 ( 2 ) ( 127 ) + 255 ( 3 ) ( 5 ) ( 17 ) + 256 ( 2 ^ 8 ) - 260 ( 2 ^ 2 ) ( 5 ) ( 13 ) - 261 ( 3 ^ 2 ) ( 29 ) + 264 ( 2 ^ 3 ) ( 3 ) ( 11 ) + 265 ( 5 ) ( 53 ) - 266 ( 2 ) ( 7 ) ( 19 ) - 267 ( 3 ) ( 89 ) - 270 ( 2 ) ( 3 ^ 3 ) ( 5 ) - 272 ( 2 ^ 4 ) ( 17 ) + 273 ( 3 ) ( 7 ) ( 13 ) - 275 ( 5 ^ 2 ) ( 11 ) + 279 ( 3 ^ 2 ) ( 31 ) + 280 ( 2 ^ 3 ) ( 5 ) ( 7 ) - 288 ( 2 ^ 5 ) ( 3 ^ 2 ) - 290 ( 2 ) ( 5 ) ( 29 ) + 292 ( 2 ^ 2 ) ( 73 ) - 294 ( 2 ) ( 3 ) ( 7 ^ 2 ) - 296 ( 2 ^ 3 ) ( 37 ) + 297 ( 3 ^ 3 ) ( 11 ) + 298 ( 2 ) ( 149 ) - 299 ( 13 ) ( 23 ) + 300 ( 2 ^ 2 ) ( 3 ) ( 5 ^ 2 ) + 301 ( 7 ) ( 43 ) - 302 ( 2 ) ( 151 ) + 304 ( 2 ^ 4 ) ( 19 ) + 306 ( 2 ) ( 3 ^ 2 ) ( 17 ) - 308 ( 2 ^ 2 ) ( 7 ) ( 11 ) + 310 ( 2 ) ( 5 ) ( 31 ) - 312 ( 2 ^ 3 ) ( 3 ) ( 13 ) + 313 ( 313 ) - 314 ( 2 ) ( 157 ) - 315 ( 3 ^ 2 ) ( 5 ) ( 7 ) + 316 ( 2 ^ 2 ) ( 79 ) - 317 ( 317 ) + 318 ( 2 ) ( 3 ) ( 53 ) - 320 ( 2 ^ 6 ) ( 5 ) + 321 ( 3 ) ( 107 ) + 322 ( 2 ) ( 7 ) ( 23 ) - 324 ( 2 ^ 2 ) ( 3 ^ 4 ) + 325 ( 5 ^ 2 ) ( 13 ) - 326 ( 2 ) ( 163 ) + 327 ( 3 ) ( 109 ) + 328 ( 2 ^ 3 ) ( 41 ) - 330 ( 2 ) ( 3 ) ( 5 ) ( 11 ) - 332 ( 2 ^ 2 ) ( 83 ) + 334 ( 2 ) ( 167 ) - 335 ( 5 ) ( 67 ) + 336 ( 2 ^ 4 ) ( 3 ) ( 7 ) - 337 ( 337 ) + 340 ( 2 ^ 2 ) ( 5 ) ( 17 ) - 344 ( 2 ^ 3 ) ( 43 ) + 346 ( 2 ) ( 173 ) - 348 ( 2 ^ 2 ) ( 3 ) ( 29 ) + 349 ( 349 ) - 350 ( 2 ) ( 5 ^ 2 ) ( 7 ) - 351 ( 3 ^ 3 ) ( 13 ) + 352 ( 2 ^ 5 ) ( 11 ) + 354 ( 2 ) ( 3 ) ( 59 ) - 356 ( 2 ^ 2 ) ( 89 ) - 357 ( 3 ) ( 7 ) ( 17 ) + 360 ( 2 ^ 3 ) ( 3 ^ 2 ) ( 5 ) + 364 ( 2 ^ 2 ) ( 7 ) ( 13 ) - 365 ( 5 ) ( 73 ) - 368 ( 2 ^ 4 ) ( 23 ) + 370 ( 2 ) ( 5 ) ( 37 ) - 371 ( 7 ) ( 53 ) + 372 ( 2 ^ 2 ) ( 3 ) ( 31 ) - 374 ( 2 ) ( 11 ) ( 17 ) - 375 ( 3 ) ( 5 ^ 3 ) + 376 ( 2 ^ 3 ) ( 47 ) - 378 ( 2 ) ( 3 ^ 3 ) ( 7 ) - 380 ( 2 ^ 2 ) ( 5 ) ( 19 ) + 381 ( 3 ) ( 127 ) - 384 ( 2 ^ 7 ) ( 3 ) + 385 ( 5 ) ( 7 ) ( 11 ) + 388 ( 2 ^ 2 ) ( 97 ) + 390 ( 2 ) ( 3 ) ( 5 ) ( 13 ) - 392 ( 2 ^ 3 ) ( 7 ^ 2 ) - 395 ( 5 ) ( 79 ) - 396 ( 2 ^ 2 ) ( 3 ^ 2 ) ( 11 ) + 399 ( 3 ) ( 7 ) ( 19 ) + 400 ( 2 ^ 4 ) ( 5 ^ 2 ) - 404 ( 2 ^ 2 ) ( 101 ) + 405 ( 3 ^ 4 ) ( 5 ) + 406 ( 2 ) ( 7 ) ( 29 ) + 408 ( 2 ^ 3 ) ( 3 ) ( 17 ) - 410 ( 2 ) ( 5 ) ( 41 ) + 412 ( 2 ^ 2 ) ( 103 ) + 415 ( 5 ) ( 83 ) - 416 ( 2 ^ 5 ) ( 13 ) + 418 ( 2 ) ( 11 ) ( 19 ) - 420 ( 2 ^ 2 ) ( 3 ) ( 5 ) ( 7 ) + 424 ( 2 ^ 3 ) ( 53 ) - 425 ( 5 ^ 2 ) ( 17 ) - 428 ( 2 ^ 2 ) ( 107 ) + 430 ( 2 ) ( 5 ) ( 43 ) + 432 ( 2 ^ 4 ) ( 3 ^ 3 ) - 434 ( 2 ) ( 7 ) ( 31 ) + 435 ( 3 ) ( 5 ) ( 29 ) - 436 ( 2 ^ 2 ) ( 109 ) - 438 ( 2 ) ( 3 ) ( 73 ) - 440 ( 2 ^ 3 ) ( 5 ) ( 11 ) + 441 ( 3 ^ 2 ) ( 7 ^ 2 ) + 442 ( 2 ) ( 13 ) ( 17 ) - 443 ( 443 ) + 444 ( 2 ^ 2 ) ( 3 ) ( 37 ) + 445 ( 5 ) ( 89 ) - 446 ( 2 ) ( 223 ) - 447 ( 3 ) ( 149 ) + 448 ( 2 ^ 6 ) ( 7 ) - 450 ( 2 ) ( 3 ^ 2 ) ( 5 ^ 2 ) - 452 ( 2 ^ 2 ) ( 113 ) + 453 ( 3 ) ( 151 ) + 454 ( 2 ) ( 227 ) - 455 ( 5 ) ( 7 ) ( 13 ) - 456 ( 2 ^ 3 ) ( 3 ) ( 19 ) + 458 ( 2 ) ( 229 ) - 459 ( 3 ^ 3 ) ( 17 ) + 460 ( 2 ^ 2 ) ( 5 ) ( 23 ) + 462 ( 2 ) ( 3 ) ( 7 ) ( 11 ) - 464 ( 2 ^ 4 ) ( 29 ) - 465 ( 3 ) ( 5 ) ( 31 ) + 468 ( 2 ^ 2 ) ( 3 ^ 2 ) ( 13 ) - 470 ( 2 ) ( 5 ) ( 47 ) + 471 ( 3 ) ( 157 ) + 472 ( 2 ^ 3 ) ( 59 ) - 473 ( 11 ) ( 43 ) - 474 ( 2 ) ( 3 ) ( 79 ) + 475 ( 5 ^ 2 ) ( 19 ) - 476 ( 2 ^ 2 ) ( 7 ) ( 17 ) - 477 ( 3 ^ 2 ) ( 53 ) + 480 ( 2 ^ 5 ) ( 3 ) ( 5 ) - 482 ( 2 ) ( 241 ) - 483 ( 3 ) ( 7 ) ( 23 ) + 484 ( 2 ^ 2 ) ( 11 ^ 2 ) - 485 ( 5 ) ( 97 ) + 486 ( 2 ) ( 3 ^ 5 ) - 488 ( 2 ^ 3 ) ( 61 ) + 489 ( 3 ) ( 163 ) + 490 ( 2 ) ( 5 ) ( 7 ^ 2 ) - 492 ( 2 ^ 2 ) ( 3 ) ( 41 ) - 494 ( 2 ) ( 13 ) ( 19 ) + 495 ( 3 ^ 2 ) ( 5 ) ( 11 ) + 496 ( 2 ^ 4 ) ( 31 ) + 498 ( 2 ) ( 3 ) ( 83 ) - 500 ( 2 ^ 2 ) ( 5 ^ 3 ) - 501 ( 3 ) ( 167 ) + 502 ( 2 ) ( 251 ) - 504 ( 2 ^ 3 ) ( 3 ^ 2 ) ( 7 ) + 505 ( 5 ) ( 101 ) - 506 ( 2 ) ( 11 ) ( 23 ) + 508 ( 2 ^ 2 ) ( 127 ) - 510 ( 2 ) ( 3 ) ( 5 ) ( 17 ) + 511 ( 7 ) ( 73 ) - 512 ( 2 ^ 9 ) + 514 ( 2 ) ( 257 ) - 515 ( 5 ) ( 103 ) + 516 ( 2 ^ 2 ) ( 3 ) ( 43 ) + 517 ( 11 ) ( 47 ) - 518 ( 2 ) ( 7 ) ( 37 ) - 519 ( 3 ) ( 173 ) + 520 ( 2 ^ 3 ) ( 5 ) ( 13 ) - 521 ( 521 ) + 522 ( 2 ) ( 3 ^ 2 ) ( 29 ) - 523 ( 523 ) + 524 ( 2 ^ 2 ) ( 131 ) + 525 ( 3 ) ( 5 ^ 2 ) ( 7 ) - 526 ( 2 ) ( 263 ) - 528 ( 2 ^ 4 ) ( 3 ) ( 11 ) - 530 ( 2 ) ( 5 ) ( 53 ) + 532 ( 2 ^ 2 ) ( 7 ) ( 19 ) + 534 ( 2 ) ( 3 ) ( 89 ) - 536 ( 2 ^ 3 ) ( 67 ) - 539 ( 7 ^ 2 ) ( 11 ) - 540 ( 2 ^ 2 ) ( 3 ^ 3 ) ( 5 ) + 544 ( 2 ^ 5 ) ( 17 ) + 545 ( 5 ) ( 109 ) - 546 ( 2 ) ( 3 ) ( 7 ) ( 13 ) - 548 ( 2 ^ 2 ) ( 137 ) + 550 ( 2 ) ( 5 ^ 2 ) ( 11 ) + 552 ( 2 ^ 3 ) ( 3 ) ( 23 ) - 555 ( 3 ) ( 5 ) ( 37 ) + 556 ( 2 ^ 2 ) ( 139 ) - 558 ( 2 ) ( 3 ^ 2 ) ( 31 ) + 559 ( 13 ) ( 43 ) - 560 ( 2 ^ 4 ) ( 5 ) ( 7 ) + 561 ( 3 ) ( 11 ) ( 17 ) + 562 ( 2 ) ( 281 ) - 564 ( 2 ^ 2 ) ( 3 ) ( 47 ) + 565 ( 5 ) ( 113 ) + 567 ( 3 ^ 4 ) ( 7 ) + 568 ( 2 ^ 3 ) ( 71 ) + 570 ( 2 ) ( 3 ) ( 5 ) ( 19 ) - 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