Ensamma Bråk – Fuska.se forum

Mezox Nivå 11 · Veteranfuskare 6 465 inlägg
medlem sedan 2004

30 mar 2013 Trådstartare #1

Som vi vet är bråk tal som skrivs som en kvot av två heltal. T.ex. 5 sjättedelar, 3 sjundedelar, en halv o.s.v.. Mer generellt kan man skriva bråk som a/b där a och b är heltal.

Låt oss kalla ett bråk för ett "ensamt bråk" om det kan skrivas som 1/b där b är ett positivt heltal. Med andra ord är alla positiva bråk med en täljare lika med 1 ensamma bråk. De ser ut som 1/1, 1/2, 1/3, 1/4, osv.

Låt oss nu välja ett ensamt bråk. Vi ställer oss frågan: Kan man alltid skriva detta ensamma bråk som en summa av två andra ensamma bråk? Om man kan, kan man göra det på flera sätt? Finns det ett sätt att konstruera en sådan summa?

Till exempel är 1/6 ett ensamt bråk. 1/6 är lika med summan av 1/10 och 1/15, men också en summa av 1/12 med sig självt. Hitta fler exempel på detta och försök hitta mönster!

♥ 3 — visa vilka

♥ Dave37, ♥ SURF, ♥ WASD

WASD Nivå 16 · Legendfuskare 11 841 inlägg
medlem sedan 2004

30 mar 2013 #2

1/(2b)+1/(2b) förstås. Att hitta två olika tal såsom 1/10+1/15 är intressantare.

Mezox Nivå 11 · Veteranfuskare 6 465 inlägg
medlem sedan 2004

31 mar 2013 Trådstartare #3

Japp :D. Finns det alltid andra sätt att skriva ett ensamt bråk?

Dave37 Nivå 13 · Mästerfuskare 6 910 inlägg
medlem sedan 2006

31 mar 2013 #4

1/a + 1/b = 1/c

(b+a)/(ba) = 1/c

c = (ba)/(b+a)

hmm... det verkar som om det finns alltid finns två sätt att skriva ensamma bråk.

http://www.wolframalpha.com/input/?i=%28floor%28b%29*floor%28a%29%29%2F%28floor%28b%29%2Bfloor%28a%29%29+%3D+37

EDIT: Jag vaknade för någon minut sen så jag orkar inte fundera på det mer just nu.

Senast ändrad: 31 mar 2013 10:24

WASD Nivå 16 · Legendfuskare 11 841 inlägg
medlem sedan 2004

31 mar 2013 #5

Jag tror då inte att det finns något annat sätt att skriva 1/1 eller 1/2 på.

Dave37 Nivå 13 · Mästerfuskare 6 910 inlägg
medlem sedan 2006

31 mar 2013 #6

1/1 går bara att skriva som 1/2+1/2 ja, 1/2 går att skriva som 1/3 + 1/6

WASD Nivå 16 · Legendfuskare 11 841 inlägg
medlem sedan 2004

31 mar 2013 #7

Jag tar och skriver ett program som brute forcear fram en lista på alla ensamma bråk och summor av andra ensamma bråk som resulterar i dem...

Dave37 Nivå 13 · Mästerfuskare 6 910 inlägg
medlem sedan 2006

31 mar 2013 #8

Nä men det måste ju finnas något relativt enkelt mönster.

t.ex.

1/2 kan skrivas som 1/3 + 1/6

1/4 kan skrivas som 1/6 + 1/12

1/8 kan skrivas som 1/12 + 1/24

1/(2n) kan skrivas som 1/(3n) + 1/(6n)

Om n är ett heltal större eller lika med 1.

för udda tal

1/3 kan skrivas som 1/4 + 1/12

1/5 kan skrivas som 1/6 + 1/30

1/7 kan skrivas som 1/8 + 1/56

1/(2n+1) kan skrivas som 1/(2+2n)+1/(2+6n+4n²)

Om n är ett heltal större eller lika med 0.

Specialfallet blir för udda tal då n är 0 och formeln reduceras till de andra lösningarna som är på formen 1/(2a) + 1/(2a) = 1/a, och i detta specialfall är a = 1.

Frågan är hur och om det går att generalisera ännu mera.

EDIT: Insåg att jag gjort en massa fel, återkommer senare!

Senast ändrad: 31 mar 2013 16:24

WASD Nivå 16 · Legendfuskare 11 841 inlägg
medlem sedan 2004

31 mar 2013 #9

1 till 100 utan 1/(2b)+1/(2b).

Exemplet "1/11 == 1/12 + 1/132" är inte med då en divisor var över 100. Kan göra mer simuleringar om så önskas.

1/2 == 1/3 + 1/6
1/3 == 1/4 + 1/12
1/4 == 1/5 + 1/20
1/4 == 1/6 + 1/12
1/5 == 1/6 + 1/30
1/6 == 1/7 + 1/42
1/6 == 1/8 + 1/24
1/6 == 1/9 + 1/18
1/6 == 1/10 + 1/15
1/7 == 1/8 + 1/56
1/8 == 1/9 + 1/72
1/8 == 1/10 + 1/40
1/8 == 1/12 + 1/24
1/9 == 1/10 + 1/90
1/9 == 1/12 + 1/36
1/10 == 1/12 + 1/60
1/10 == 1/14 + 1/35
1/10 == 1/15 + 1/30
1/12 == 1/14 + 1/84
1/12 == 1/15 + 1/60
1/12 == 1/16 + 1/48
1/12 == 1/18 + 1/36
1/12 == 1/20 + 1/30
1/12 == 1/21 + 1/28
1/14 == 1/18 + 1/63
1/14 == 1/21 + 1/42
1/15 == 1/18 + 1/90
1/15 == 1/20 + 1/60
1/15 == 1/24 + 1/40
1/16 == 1/20 + 1/80
1/16 == 1/24 + 1/48
1/18 == 1/22 + 1/99
1/18 == 1/24 + 1/72
1/18 == 1/27 + 1/54
1/18 == 1/30 + 1/45
1/20 == 1/25 + 1/100
1/20 == 1/28 + 1/70
1/20 == 1/30 + 1/60
1/20 == 1/36 + 1/45
1/21 == 1/28 + 1/84
1/21 == 1/30 + 1/70
1/22 == 1/33 + 1/66
1/24 == 1/32 + 1/96
1/24 == 1/33 + 1/88
1/24 == 1/36 + 1/72
1/24 == 1/40 + 1/60
1/24 == 1/42 + 1/56
1/26 == 1/39 + 1/78
1/28 == 1/42 + 1/84
1/28 == 1/44 + 1/77
1/30 == 1/45 + 1/90
1/30 == 1/48 + 1/80
1/30 == 1/50 + 1/75
1/30 == 1/55 + 1/66
1/32 == 1/48 + 1/96
1/35 == 1/60 + 1/84
1/36 == 1/60 + 1/90
1/36 == 1/63 + 1/84
1/40 == 1/72 + 1/90
1/42 == 1/78 + 1/91

Senast ändrad: 31 mar 2013 16:29

Dave37 Nivå 13 · Mästerfuskare 6 910 inlägg
medlem sedan 2006

31 mar 2013 #10

Fast med wolfram alphas hjälp hittade jag följande bråk för t.ex. 1/8:

1/9 + 1/72

1/10 + 1/40

1/12 + 1/24

1/16 + 1/16

Jag ska kolla närmare in på sifferföljden 9, 10, 12, 16, 24, 40, 72 och jämföra den med de sifferföljder jag lär få andra ensamma bråk sen.

EDIT: såg att du också fått fram dem nu. WASD, gör om simuleringen fast med bråken som som är 1/(2a) + 1/(2a). Eller ja, plocka ut talföljderna för varje nämnare. Men denna gången, gör det ner till 1/10 och ta med alla bråk som programmet hittar, även om de är mindre än 1/100.

Senast ändrad: 31 mar 2013 17:51

WASD Nivå 16 · Legendfuskare 11 841 inlägg
medlem sedan 2004

31 mar 2013 #11

Nämnare 2-10 med summor av ensamma bråk ner till 1/10000

Edit: Bortsett från 1/b == 1/(2b)+1/(2b)

1/2 == 1/3 + 1/6
1/3 == 1/4 + 1/12
1/4 == 1/5 + 1/20
1/4 == 1/6 + 1/12
1/5 == 1/6 + 1/30
1/6 == 1/7 + 1/42
1/6 == 1/8 + 1/24
1/6 == 1/9 + 1/18
1/6 == 1/10 + 1/15
1/7 == 1/8 + 1/56
1/8 == 1/9 + 1/72
1/8 == 1/10 + 1/40
1/8 == 1/12 + 1/24
1/9 == 1/10 + 1/90
1/9 == 1/12 + 1/36
1/10 == 1/11 + 1/110
1/10 == 1/12 + 1/60
1/10 == 1/14 + 1/35
1/10 == 1/15 + 1/30

Endast det fetmarkerade har ett tal över 100 och jag körde som sagt försök ner till 1/10000 så det finns nästan garanterat inget under det. Denna listan påstår jag således är alla ensamma bråk som kan bilda de ensamma bråken 1/2 till 1/10.

Ett mönster jag ser i 1/a == 1/b + 1/c är att c oftast är jämt delbart med b.

Senast ändrad: 31 mar 2013 18:44

Dave37 Nivå 13 · Mästerfuskare 6 910 inlägg
medlem sedan 2006

31 mar 2013 #12

ok så då har vi följande:

2:

3, 4, 6

3:

4, 6, 12

4:

5, 6, 8, 12, 20

5:

6, 10, 30

6:

7, 8, 9, 12, 18, 24, 42

7:

8, 14, 58

8:

9, 10, 12, 16, 24, 40, 72

9:

10, 12, 18, 36, 90

10:

11, 12, 14, 15, 20, 30, 35, 60, 110

Hmm... Jag ser inget mönster direkt... förutom att det verkar som om primtalen bara kan delas upp på två sätt. WASD, testa att köra med bara primtal, typ alla primtal upp till 100.

WASD Nivå 16 · Legendfuskare 11 841 inlägg
medlem sedan 2004

31 mar 2013 #13

Körde en massa primtal upp till 500 med försök att summera ner till 1/1000 och de flesta kunde bildas på många olika sätt.

1/479 == 1/921 + 1/998
1/479 == 1/922 + 1/997
1/479 == 1/928 + 1/990
1/479 == 1/936 + 1/981
1/479 == 1/937 + 1/980
1/479 == 1/952 + 1/964
1/479 == 1/953 + 1/963
1/479 == 1/954 + 1/962
1/479 == 1/955 + 1/961
1/479 == 1/956 + 1/960
1/479 == 1/957 + 1/959

Mezox har du skype? Jag och davve snackar där.

Här är alla resultat. Första primtalsensammabråket att ha mer än en möjlighet är 103
Edit: Och de primtalen som inte står med uppfylls endast av 1/(2b)+1/(2b).

Edit: På grund av avrundningsfel är denna listan inkorrekt. Ensamma bråk av primtal verkar bara ha en lösning utöver 1/(2b)+1/(2b)

1/3 == 1/4 + 1/12
1/5 == 1/6 + 1/30
1/7 == 1/8 + 1/56
1/11 == 1/12 + 1/132
1/13 == 1/14 + 1/182
1/17 == 1/18 + 1/306
1/19 == 1/20 + 1/380
1/23 == 1/24 + 1/552
1/29 == 1/30 + 1/870
1/31 == 1/32 + 1/992
1/101 == 1/113 + 1/951
1/103 == 1/115 + 1/987
1/103 == 1/116 + 1/919
1/109 == 1/124 + 1/901
1/109 == 1/127 + 1/769
1/113 == 1/129 + 1/911
1/113 == 1/132 + 1/785
1/113 == 1/134 + 1/721
1/127 == 1/148 + 1/895
1/127 == 1/151 + 1/799
1/127 == 1/155 + 1/703
1/127 == 1/159 + 1/631
1/127 == 1/163 + 1/575
1/127 == 1/169 + 1/511
1/127 == 1/175 + 1/463
1/131 == 1/151 + 1/989
1/131 == 1/153 + 1/911
1/131 == 1/155 + 1/846
1/131 == 1/157 + 1/791
1/131 == 1/161 + 1/703
1/131 == 1/164 + 1/651
1/131 == 1/170 + 1/571
1/131 == 1/171 + 1/560
1/131 == 1/175 + 1/521
1/131 == 1/183 + 1/461
1/131 == 1/186 + 1/443
1/131 == 1/191 + 1/417
1/131 == 1/196 + 1/395
1/131 == 1/197 + 1/391
1/137 == 1/160 + 1/953
1/137 == 1/161 + 1/919
1/137 == 1/171 + 1/689
1/137 == 1/183 + 1/545
1/137 == 1/185 + 1/528
1/137 == 1/188 + 1/505
1/137 == 1/205 + 1/413
1/137 == 1/206 + 1/409
1/137 == 1/229 + 1/341
1/137 == 1/239 + 1/321
1/137 == 1/273 + 1/275
1/139 == 1/162 + 1/979
1/139 == 1/163 + 1/944
1/139 == 1/167 + 1/829
1/139 == 1/169 + 1/783
1/139 == 1/174 + 1/691
1/139 == 1/179 + 1/622
1/139 == 1/181 + 1/599
1/139 == 1/185 + 1/559
1/139 == 1/195 + 1/484
1/139 == 1/199 + 1/461
1/139 == 1/208 + 1/419
1/139 == 1/209 + 1/415
1/139 == 1/223 + 1/369
1/139 == 1/231 + 1/349
1/139 == 1/244 + 1/323
1/139 == 1/254 + 1/307
1/139 == 1/259 + 1/300
1/139 == 1/277 + 1/279
1/149 == 1/179 + 1/889
1/149 == 1/183 + 1/802
1/149 == 1/186 + 1/749
1/149 == 1/189 + 1/704
1/149 == 1/199 + 1/593
1/149 == 1/209 + 1/519
1/149 == 1/223 + 1/449
1/149 == 1/224 + 1/445
1/149 == 1/249 + 1/371
1/149 == 1/260 + 1/349
1/149 == 1/269 + 1/334
1/149 == 1/297 + 1/299
1/151 == 1/181 + 1/911
1/151 == 1/184 + 1/842
1/151 == 1/189 + 1/751
1/151 == 1/191 + 1/721
1/151 == 1/199 + 1/626
1/151 == 1/201 + 1/607
1/151 == 1/208 + 1/551
1/151 == 1/211 + 1/531
1/151 == 1/226 + 1/455
1/151 == 1/227 + 1/451
1/151 == 1/231 + 1/436
1/151 == 1/246 + 1/391
1/151 == 1/251 + 1/379
1/151 == 1/265 + 1/351
1/151 == 1/271 + 1/341
1/151 == 1/301 + 1/303
1/157 == 1/190 + 1/904
1/157 == 1/191 + 1/882
1/157 == 1/196 + 1/789
1/157 == 1/206 + 1/660
1/157 == 1/207 + 1/650
1/157 == 1/209 + 1/631
1/157 == 1/215 + 1/582
1/157 == 1/235 + 1/473
1/157 == 1/236 + 1/469
1/157 == 1/242 + 1/447
1/157 == 1/261 + 1/394
1/157 == 1/302 + 1/327
1/157 == 1/313 + 1/315
1/163 == 1/199 + 1/901
1/163 == 1/204 + 1/811
1/163 == 1/214 + 1/684
1/163 == 1/217 + 1/655
1/163 == 1/235 + 1/532
1/163 == 1/244 + 1/491
1/163 == 1/245 + 1/487
1/163 == 1/271 + 1/409
1/163 == 1/286 + 1/379
1/163 == 1/325 + 1/327
1/167 == 1/205 + 1/901
1/167 == 1/209 + 1/831
1/167 == 1/215 + 1/748
1/167 == 1/223 + 1/665
1/167 == 1/246 + 1/520
1/167 == 1/250 + 1/503
1/167 == 1/251 + 1/499
1/167 == 1/279 + 1/416
1/167 == 1/333 + 1/335
1/173 == 1/214 + 1/903
1/173 == 1/216 + 1/869
1/173 == 1/231 + 1/689
1/173 == 1/246 + 1/583
1/173 == 1/255 + 1/538
1/173 == 1/259 + 1/521
1/173 == 1/260 + 1/517
1/173 == 1/289 + 1/431
1/173 == 1/302 + 1/405
1/173 == 1/319 + 1/378
1/173 == 1/345 + 1/347
1/179 == 1/219 + 1/980
1/179 == 1/224 + 1/891
1/179 == 1/239 + 1/713
1/179 == 1/251 + 1/624
1/179 == 1/253 + 1/612
1/179 == 1/268 + 1/539
1/179 == 1/269 + 1/535
1/179 == 1/299 + 1/446
1/179 == 1/340 + 1/378
1/179 == 1/357 + 1/359
1/181 == 1/221 + 1/1000
1/181 == 1/222 + 1/980
1/181 == 1/223 + 1/961
1/181 == 1/226 + 1/909
1/181 == 1/228 + 1/878
1/181 == 1/233 + 1/811
1/181 == 1/237 + 1/766
1/181 == 1/241 + 1/727
1/181 == 1/244 + 1/701
1/181 == 1/246 + 1/685
1/181 == 1/248 + 1/670
1/181 == 1/251 + 1/649
1/181 == 1/253 + 1/636
1/181 == 1/259 + 1/601
1/181 == 1/265 + 1/571
1/181 == 1/271 + 1/545
1/181 == 1/272 + 1/541
1/181 == 1/285 + 1/496
1/181 == 1/286 + 1/493
1/181 == 1/298 + 1/461
1/181 == 1/301 + 1/454
1/181 == 1/307 + 1/441
1/181 == 1/311 + 1/433
1/181 == 1/321 + 1/415
1/181 == 1/337 + 1/391
1/181 == 1/344 + 1/382
1/181 == 1/349 + 1/376
1/181 == 1/361 + 1/363
1/191 == 1/237 + 1/984
1/191 == 1/239 + 1/951
1/191 == 1/248 + 1/831
1/191 == 1/249 + 1/820
1/191 == 1/251 + 1/799
1/191 == 1/252 + 1/789
1/191 == 1/255 + 1/761
1/191 == 1/265 + 1/684
1/191 == 1/267 + 1/671
1/191 == 1/271 + 1/647
1/191 == 1/286 + 1/575
1/191 == 1/287 + 1/571
1/191 == 1/305 + 1/511
1/191 == 1/311 + 1/495
1/191 == 1/319 + 1/476
1/191 == 1/343 + 1/431
1/191 == 1/351 + 1/419
1/191 == 1/381 + 1/383
1/193 == 1/241 + 1/969
1/193 == 1/243 + 1/938
1/193 == 1/257 + 1/775
1/193 == 1/260 + 1/749
1/193 == 1/289 + 1/581
1/193 == 1/290 + 1/577
1/193 == 1/312 + 1/506
1/193 == 1/318 + 1/491
1/193 == 1/321 + 1/484
1/193 == 1/342 + 1/443
1/193 == 1/385 + 1/387
1/197 == 1/246 + 1/989
1/197 == 1/248 + 1/958
1/197 == 1/253 + 1/890
1/197 == 1/259 + 1/823
1/197 == 1/260 + 1/813
1/197 == 1/262 + 1/794
1/197 == 1/263 + 1/785
1/197 == 1/269 + 1/736
1/197 == 1/274 + 1/701
1/197 == 1/281 + 1/659
1/197 == 1/285 + 1/638
1/197 == 1/295 + 1/593
1/197 == 1/296 + 1/589
1/197 == 1/321 + 1/510
1/197 == 1/323 + 1/505
1/197 == 1/329 + 1/491
1/197 == 1/344 + 1/461
1/197 == 1/348 + 1/454
1/197 == 1/351 + 1/449
1/197 == 1/365 + 1/428
1/197 == 1/393 + 1/395
1/199 == 1/249 + 1/991
1/199 == 1/254 + 1/919
1/199 == 1/259 + 1/859
1/199 == 1/265 + 1/799
1/199 == 1/266 + 1/790
1/199 == 1/271 + 1/749
1/199 == 1/274 + 1/727
1/199 == 1/279 + 1/694
1/199 == 1/287 + 1/649
1/199 == 1/289 + 1/639
1/199 == 1/298 + 1/599
1/199 == 1/299 + 1/595
1/199 == 1/309 + 1/559
1/199 == 1/319 + 1/529
1/199 == 1/328 + 1/506
1/199 == 1/331 + 1/499
1/199 == 1/343 + 1/474
1/199 == 1/349 + 1/463
1/199 == 1/364 + 1/439
1/199 == 1/375 + 1/424
1/199 == 1/379 + 1/419
1/199 == 1/397 + 1/399
1/211 == 1/268 + 1/992
1/211 == 1/271 + 1/953
1/211 == 1/273 + 1/929
1/211 == 1/276 + 1/896
1/211 == 1/281 + 1/847
1/211 == 1/282 + 1/838
1/211 == 1/295 + 1/741
1/211 == 1/308 + 1/670
1/211 == 1/316 + 1/635
1/211 == 1/317 + 1/631
1/211 == 1/324 + 1/605
1/211 == 1/331 + 1/582
1/211 == 1/351 + 1/529
1/211 == 1/364 + 1/502
1/211 == 1/370 + 1/491
1/211 == 1/379 + 1/476
1/211 == 1/408 + 1/437
1/211 == 1/421 + 1/423
1/223 == 1/287 + 1/1000
1/223 == 1/288 + 1/988
1/223 == 1/297 + 1/895
1/223 == 1/298 + 1/886
1/223 == 1/304 + 1/837
1/223 == 1/307 + 1/815
1/223 == 1/308 + 1/808
1/223 == 1/317 + 1/752
1/223 == 1/319 + 1/741
1/223 == 1/334 + 1/671
1/223 == 1/335 + 1/667
1/223 == 1/340 + 1/648
1/223 == 1/344 + 1/634
1/223 == 1/360 + 1/586
1/223 == 1/371 + 1/559
1/223 == 1/376 + 1/548
1/223 == 1/391 + 1/519
1/223 == 1/415 + 1/482
1/223 == 1/418 + 1/478
1/223 == 1/444 + 1/448
1/223 == 1/445 + 1/447
1/227 == 1/294 + 1/996
1/227 == 1/302 + 1/914
1/227 == 1/303 + 1/905
1/227 == 1/316 + 1/806
1/227 == 1/340 + 1/683
1/227 == 1/341 + 1/679
1/227 == 1/346 + 1/660
1/227 == 1/353 + 1/636
1/227 == 1/379 + 1/566
1/227 == 1/420 + 1/494
1/227 == 1/452 + 1/456
1/227 == 1/453 + 1/455
1/229 == 1/298 + 1/989
1/229 == 1/305 + 1/919
1/229 == 1/306 + 1/910
1/229 == 1/314 + 1/846
1/229 == 1/321 + 1/799
1/229 == 1/324 + 1/781
1/229 == 1/343 + 1/689
1/229 == 1/344 + 1/685
1/229 == 1/349 + 1/666
1/229 == 1/367 + 1/609
1/229 == 1/381 + 1/574
1/229 == 1/386 + 1/563
1/229 == 1/396 + 1/543
1/229 == 1/413 + 1/514
1/229 == 1/419 + 1/505
1/229 == 1/456 + 1/460
1/229 == 1/457 + 1/459
1/233 == 1/305 + 1/987
1/233 == 1/310 + 1/938
1/233 == 1/311 + 1/929
1/233 == 1/315 + 1/895
1/233 == 1/320 + 1/857
1/233 == 1/322 + 1/843
1/233 == 1/331 + 1/787
1/233 == 1/337 + 1/755
1/233 == 1/338 + 1/750
1/233 == 1/349 + 1/701
1/233 == 1/350 + 1/697
1/233 == 1/355 + 1/678
1/233 == 1/374 + 1/618
1/233 == 1/377 + 1/610
1/233 == 1/389 + 1/581
1/233 == 1/397 + 1/564
1/233 == 1/398 + 1/562
1/233 == 1/407 + 1/545
1/233 == 1/411 + 1/538
1/233 == 1/429 + 1/510
1/233 == 1/441 + 1/494
1/233 == 1/464 + 1/468
1/233 == 1/465 + 1/467
1/239 == 1/318 + 1/962
1/239 == 1/319 + 1/953
1/239 == 1/323 + 1/919
1/239 == 1/324 + 1/911
1/239 == 1/335 + 1/834
1/239 == 1/338 + 1/816
1/239 == 1/341 + 1/799
1/239 == 1/344 + 1/783
1/239 == 1/348 + 1/763
1/239 == 1/351 + 1/749
1/239 == 1/358 + 1/719
1/239 == 1/359 + 1/715
1/239 == 1/364 + 1/696
1/239 == 1/370 + 1/675
1/239 == 1/375 + 1/659
1/239 == 1/379 + 1/647
1/239 == 1/399 + 1/596
1/239 == 1/407 + 1/579
1/239 == 1/408 + 1/577
1/239 == 1/409 + 1/575
1/239 == 1/443 + 1/519
1/239 == 1/449 + 1/511
1/239 == 1/457 + 1/501
1/239 == 1/463 + 1/494
1/239 == 1/476 + 1/480
1/239 == 1/477 + 1/479
1/241 == 1/321 + 1/967
1/241 == 1/322 + 1/958
1/241 == 1/329 + 1/901
1/241 == 1/337 + 1/846
1/241 == 1/342 + 1/816
1/241 == 1/351 + 1/769
1/241 == 1/354 + 1/755
1/241 == 1/356 + 1/746
1/241 == 1/361 + 1/725
1/241 == 1/362 + 1/721
1/241 == 1/367 + 1/702
1/241 == 1/373 + 1/681
1/241 == 1/383 + 1/650
1/241 == 1/401 + 1/604
1/241 == 1/406 + 1/593
1/241 == 1/417 + 1/571
1/241 == 1/461 + 1/505
1/241 == 1/467 + 1/498
1/241 == 1/480 + 1/484
1/241 == 1/481 + 1/483
1/251 == 1/339 + 1/967
1/251 == 1/341 + 1/951
1/251 == 1/351 + 1/881
1/251 == 1/356 + 1/851
1/251 == 1/360 + 1/829
1/251 == 1/371 + 1/776
1/251 == 1/376 + 1/755
1/251 == 1/377 + 1/751
1/251 == 1/391 + 1/701
1/251 == 1/401 + 1/671
1/251 == 1/419 + 1/626
1/251 == 1/426 + 1/611
1/251 == 1/431 + 1/601
1/251 == 1/451 + 1/566
1/251 == 1/461 + 1/551
1/251 == 1/466 + 1/544
1/251 == 1/469 + 1/540
1/251 == 1/476 + 1/531
1/251 == 1/500 + 1/504
1/251 == 1/501 + 1/503
1/257 == 1/349 + 1/975
1/257 == 1/353 + 1/945
1/257 == 1/354 + 1/938
1/257 == 1/355 + 1/931
1/257 == 1/358 + 1/911
1/257 == 1/362 + 1/886
1/257 == 1/366 + 1/863
1/257 == 1/368 + 1/852
1/257 == 1/376 + 1/812
1/257 == 1/380 + 1/794
1/257 == 1/385 + 1/773
1/257 == 1/386 + 1/769
1/257 == 1/415 + 1/675
1/257 == 1/429 + 1/641
1/257 == 1/436 + 1/626
1/257 == 1/442 + 1/614
1/257 == 1/449 + 1/601
1/257 == 1/453 + 1/594
1/257 == 1/459 + 1/584
1/257 == 1/466 + 1/573
1/257 == 1/468 + 1/570
1/257 == 1/475 + 1/560
1/257 == 1/512 + 1/516
1/257 == 1/513 + 1/515
1/263 == 1/358 + 1/991
1/263 == 1/360 + 1/976
1/263 == 1/385 + 1/830
1/263 == 1/389 + 1/812
1/263 == 1/394 + 1/791
1/263 == 1/395 + 1/787
1/263 == 1/408 + 1/740
1/263 == 1/422 + 1/698
1/263 == 1/425 + 1/690
1/263 == 1/439 + 1/656
1/263 == 1/446 + 1/641
1/263 == 1/452 + 1/629
1/263 == 1/484 + 1/576
1/263 == 1/504 + 1/550
1/263 == 1/524 + 1/528
1/263 == 1/525 + 1/527
1/269 == 1/368 + 1/1000
1/269 == 1/377 + 1/939
1/269 == 1/381 + 1/915
1/269 == 1/388 + 1/877
1/269 == 1/389 + 1/872
1/269 == 1/390 + 1/867
1/269 == 1/398 + 1/830
1/269 == 1/403 + 1/809
1/269 == 1/404 + 1/805
1/269 == 1/412 + 1/775
1/269 == 1/414 + 1/768
1/269 == 1/422 + 1/742
1/269 == 1/427 + 1/727
1/269 == 1/449 + 1/671
1/269 == 1/455 + 1/658
1/269 == 1/456 + 1/656
1/269 == 1/463 + 1/642
1/269 == 1/470 + 1/629
1/269 == 1/474 + 1/622
1/269 == 1/485 + 1/604
1/269 == 1/498 + 1/585
1/269 == 1/511 + 1/568
1/269 == 1/522 + 1/555
1/269 == 1/536 + 1/540
1/269 == 1/537 + 1/539
1/271 == 1/373 + 1/991
1/271 == 1/374 + 1/984
1/271 == 1/379 + 1/951
1/271 == 1/384 + 1/921
1/271 == 1/391 + 1/883
1/271 == 1/392 + 1/878
1/271 == 1/393 + 1/873
1/271 == 1/401 + 1/836
1/271 == 1/406 + 1/815
1/271 == 1/407 + 1/811
1/271 == 1/415 + 1/781
1/271 == 1/417 + 1/774
1/271 == 1/424 + 1/751
1/271 == 1/431 + 1/730
1/271 == 1/441 + 1/703
1/271 == 1/443 + 1/698
1/271 == 1/451 + 1/679
1/271 == 1/456 + 1/668
1/271 == 1/475 + 1/631
1/271 == 1/487 + 1/611
1/271 == 1/511 + 1/577
1/271 == 1/515 + 1/572
1/271 == 1/526 + 1/559
1/271 == 1/540 + 1/544
1/271 == 1/541 + 1/543
1/277 == 1/384 + 1/994
1/277 == 1/386 + 1/981
1/277 == 1/389 + 1/962
1/277 == 1/390 + 1/956
1/277 == 1/391 + 1/950
1/277 == 1/399 + 1/906
1/277 == 1/403 + 1/886
1/277 == 1/414 + 1/837
1/277 == 1/415 + 1/833
1/277 == 1/416 + 1/829
1/277 == 1/417 + 1/825
1/277 == 1/424 + 1/799
1/277 == 1/426 + 1/792
1/277 == 1/432 + 1/772
1/277 == 1/442 + 1/742
1/277 == 1/446 + 1/731
1/277 == 1/451 + 1/718
1/277 == 1/453 + 1/713
1/277 == 1/461 + 1/694
1/277 == 1/466 + 1/683
1/277 == 1/480 + 1/655
1/277 == 1/495 + 1/629
1/277 == 1/502 + 1/618
1/277 == 1/504 + 1/615
1/277 == 1/538 + 1/571
1/277 == 1/552 + 1/556
1/277 == 1/553 + 1/555
1/281 == 1/393 + 1/986
1/281 == 1/401 + 1/939
1/281 == 1/404 + 1/923
1/281 == 1/420 + 1/849
1/281 == 1/421 + 1/845
1/281 == 1/422 + 1/841
1/281 == 1/423 + 1/837
1/281 == 1/430 + 1/811
1/281 == 1/449 + 1/751
1/281 == 1/469 + 1/701
1/281 == 1/475 + 1/688
1/281 == 1/484 + 1/670
1/281 == 1/491 + 1/657
1/281 == 1/495 + 1/650
1/281 == 1/516 + 1/617
1/281 == 1/521 + 1/610
1/281 == 1/527 + 1/602
1/281 == 1/560 + 1/564
1/281 == 1/561 + 1/563
1/283 == 1/395 + 1/998
1/283 == 1/402 + 1/956
1/283 == 1/413 + 1/899
1/283 == 1/423 + 1/855
1/283 == 1/424 + 1/851
1/283 == 1/425 + 1/847
1/283 == 1/426 + 1/843
1/283 == 1/437 + 1/803
1/283 == 1/459 + 1/738
1/283 == 1/465 + 1/723
1/283 == 1/471 + 1/709
1/283 == 1/476 + 1/698
1/283 == 1/491 + 1/668
1/283 == 1/496 + 1/659
1/283 == 1/503 + 1/647
1/283 == 1/543 + 1/591
1/283 == 1/563 + 1/569
1/283 == 1/564 + 1/568
1/283 == 1/565 + 1/567
1/293 == 1/416 + 1/991
1/293 == 1/420 + 1/969
1/293 == 1/428 + 1/929
1/293 == 1/438 + 1/885
1/293 == 1/439 + 1/881
1/293 == 1/440 + 1/877
1/293 == 1/441 + 1/873
1/293 == 1/452 + 1/833
1/293 == 1/461 + 1/804
1/293 == 1/462 + 1/801
1/293 == 1/463 + 1/798
1/293 == 1/464 + 1/795
1/293 == 1/470 + 1/778
1/293 == 1/478 + 1/757
1/293 == 1/489 + 1/731
1/293 == 1/495 + 1/718
1/293 == 1/506 + 1/696
1/293 == 1/512 + 1/685
1/293 == 1/516 + 1/678
1/293 == 1/525 + 1/663
1/293 == 1/539 + 1/642
1/293 == 1/544 + 1/635
1/293 == 1/547 + 1/631
1/293 == 1/583 + 1/589
1/293 == 1/584 + 1/588
1/293 == 1/585 + 1/587
1/307 == 1/443 + 1/1000
1/307 == 1/444 + 1/995
1/307 == 1/445 + 1/990
1/307 == 1/446 + 1/985
1/307 == 1/450 + 1/966
1/307 == 1/452 + 1/957
1/307 == 1/459 + 1/927
1/307 == 1/460 + 1/923
1/307 == 1/461 + 1/919
1/307 == 1/462 + 1/915
1/307 == 1/467 + 1/896
1/307 == 1/475 + 1/868
1/307 == 1/479 + 1/855
1/307 == 1/490 + 1/822
1/307 == 1/494 + 1/811
1/307 == 1/497 + 1/803
1/307 == 1/505 + 1/783
1/307 == 1/511 + 1/769
1/307 == 1/516 + 1/758
1/307 == 1/533 + 1/724
1/307 == 1/538 + 1/715
1/307 == 1/545 + 1/703
1/307 == 1/555 + 1/687
1/307 == 1/557 + 1/684
1/307 == 1/559 + 1/681
1/307 == 1/571 + 1/664
1/307 == 1/574 + 1/660
1/307 == 1/581 + 1/651
1/307 == 1/585 + 1/646
1/307 == 1/590 + 1/640
1/307 == 1/597 + 1/632
1/307 == 1/611 + 1/617
1/307 == 1/612 + 1/616
1/307 == 1/613 + 1/615
1/311 == 1/452 + 1/997
1/311 == 1/456 + 1/978
1/311 == 1/458 + 1/969
1/311 == 1/465 + 1/939
1/311 == 1/466 + 1/935
1/311 == 1/467 + 1/931
1/311 == 1/468 + 1/927
1/311 == 1/473 + 1/908
1/311 == 1/481 + 1/880
1/311 == 1/497 + 1/831
1/311 == 1/501 + 1/820
1/311 == 1/506 + 1/807
1/311 == 1/508 + 1/802
1/311 == 1/510 + 1/797
1/311 == 1/519 + 1/776
1/311 == 1/525 + 1/763
1/311 == 1/537 + 1/739
1/311 == 1/551 + 1/714
1/311 == 1/554 + 1/709
1/311 == 1/559 + 1/701
1/311 == 1/571 + 1/683
1/311 == 1/576 + 1/676
1/311 == 1/585 + 1/664
1/311 == 1/593 + 1/654
1/311 == 1/598 + 1/648
1/311 == 1/605 + 1/640
1/311 == 1/619 + 1/625
1/311 == 1/620 + 1/624
1/311 == 1/621 + 1/623
1/313 == 1/456 + 1/998
1/313 == 1/459 + 1/984
1/313 == 1/461 + 1/975
1/313 == 1/468 + 1/945
1/313 == 1/469 + 1/941
1/313 == 1/470 + 1/937
1/313 == 1/471 + 1/933
1/313 == 1/476 + 1/914
1/313 == 1/487 + 1/876
1/313 == 1/507 + 1/818
1/313 == 1/515 + 1/798
1/313 == 1/521 + 1/784
1/313 == 1/526 + 1/773
1/313 == 1/543 + 1/739
1/313 == 1/561 + 1/708
1/313 == 1/589 + 1/668
1/313 == 1/597 + 1/658
1/313 == 1/602 + 1/652
1/313 == 1/609 + 1/644
1/313 == 1/623 + 1/629
1/313 == 1/624 + 1/628
1/313 == 1/625 + 1/627
1/317 == 1/465 + 1/996
1/317 == 1/467 + 1/987
1/317 == 1/474 + 1/957
1/317 == 1/475 + 1/953
1/317 == 1/476 + 1/949
1/317 == 1/477 + 1/945
1/317 == 1/482 + 1/926
1/317 == 1/493 + 1/888
1/317 == 1/511 + 1/835
1/317 == 1/516 + 1/822
1/317 == 1/518 + 1/817
1/317 == 1/520 + 1/812
1/317 == 1/529 + 1/791
1/317 == 1/535 + 1/778
1/317 == 1/548 + 1/752
1/317 == 1/554 + 1/741
1/317 == 1/558 + 1/734
1/317 == 1/567 + 1/719
1/317 == 1/574 + 1/708
1/317 == 1/576 + 1/705
1/317 == 1/578 + 1/702
1/317 == 1/585 + 1/692
1/317 == 1/610 + 1/660
1/317 == 1/617 + 1/652
1/317 == 1/631 + 1/637
1/317 == 1/632 + 1/636
1/317 == 1/633 + 1/635
1/331 == 1/495 + 1/999
1/331 == 1/496 + 1/995
1/331 == 1/497 + 1/991
1/331 == 1/498 + 1/987
1/331 == 1/503 + 1/968
1/331 == 1/506 + 1/957
1/331 == 1/508 + 1/950
1/331 == 1/510 + 1/943
1/331 == 1/513 + 1/933
1/331 == 1/517 + 1/920
1/331 == 1/527 + 1/890
1/331 == 1/535 + 1/868
1/331 == 1/545 + 1/843
1/331 == 1/551 + 1/829
1/331 == 1/556 + 1/818
1/331 == 1/575 + 1/780
1/331 == 1/580 + 1/771
1/331 == 1/584 + 1/764
1/331 == 1/587 + 1/759
1/331 == 1/590 + 1/754
1/331 == 1/595 + 1/746
1/331 == 1/607 + 1/728
1/331 == 1/632 + 1/695
1/331 == 1/637 + 1/689
1/331 == 1/644 + 1/681
1/331 == 1/659 + 1/665
1/331 == 1/660 + 1/664
1/331 == 1/661 + 1/663
1/337 == 1/512 + 1/986
1/337 == 1/515 + 1/975
1/337 == 1/517 + 1/968
1/337 == 1/519 + 1/961
1/337 == 1/538 + 1/902
1/337 == 1/542 + 1/891
1/337 == 1/545 + 1/883
1/337 == 1/555 + 1/858
1/337 == 1/561 + 1/844
1/337 == 1/566 + 1/833
1/337 == 1/585 + 1/795
1/337 == 1/603 + 1/764
1/337 == 1/610 + 1/753
1/337 == 1/612 + 1/750
1/337 == 1/614 + 1/747
1/337 == 1/626 + 1/730
1/337 == 1/632 + 1/722
1/337 == 1/649 + 1/701
1/337 == 1/656 + 1/693
1/337 == 1/671 + 1/677
1/337 == 1/672 + 1/676
1/337 == 1/673 + 1/675
1/347 == 1/536 + 1/984
1/347 == 1/546 + 1/952
1/347 == 1/547 + 1/949
1/347 == 1/548 + 1/946
1/347 == 1/549 + 1/943
1/347 == 1/559 + 1/915
1/347 == 1/562 + 1/907
1/347 == 1/570 + 1/887
1/347 == 1/579 + 1/866
1/347 == 1/584 + 1/855
1/347 == 1/601 + 1/821
1/347 == 1/614 + 1/798
1/347 == 1/617 + 1/793
1/347 == 1/620 + 1/788
1/347 == 1/627 + 1/777
1/347 == 1/629 + 1/774
1/347 == 1/631 + 1/771
1/347 == 1/633 + 1/768
1/347 == 1/640 + 1/758
1/347 == 1/648 + 1/747
1/347 == 1/663 + 1/728
1/347 == 1/676 + 1/713
1/347 == 1/690 + 1/698
1/347 == 1/691 + 1/697
1/347 == 1/692 + 1/696
1/347 == 1/693 + 1/695
1/349 == 1/539 + 1/990
1/349 == 1/542 + 1/980
1/349 == 1/548 + 1/961
1/349 == 1/549 + 1/958
1/349 == 1/550 + 1/955
1/349 == 1/551 + 1/952
1/349 == 1/552 + 1/949
1/349 == 1/553 + 1/946
1/349 == 1/559 + 1/929
1/349 == 1/575 + 1/888
1/349 == 1/581 + 1/874
1/349 == 1/586 + 1/863
1/349 == 1/606 + 1/823
1/349 == 1/629 + 1/784
1/349 == 1/639 + 1/769
1/349 == 1/649 + 1/755
1/349 == 1/652 + 1/751
1/349 == 1/655 + 1/747
1/349 == 1/667 + 1/732
1/349 == 1/680 + 1/717
1/349 == 1/694 + 1/702
1/349 == 1/695 + 1/701
1/349 == 1/696 + 1/700
1/349 == 1/697 + 1/699
1/353 == 1/548 + 1/992
1/353 == 1/553 + 1/976
1/353 == 1/554 + 1/973
1/353 == 1/560 + 1/955
1/353 == 1/561 + 1/952
1/353 == 1/566 + 1/938
1/353 == 1/572 + 1/922
1/353 == 1/580 + 1/902
1/353 == 1/589 + 1/881
1/353 == 1/594 + 1/870
1/353 == 1/611 + 1/836
1/353 == 1/617 + 1/825
1/353 == 1/621 + 1/818
1/353 == 1/633 + 1/798
1/353 == 1/649 + 1/774
1/353 == 1/654 + 1/767
1/353 == 1/663 + 1/755
1/353 == 1/675 + 1/740
1/353 == 1/688 + 1/725
1/353 == 1/702 + 1/710
1/353 == 1/703 + 1/709
1/353 == 1/704 + 1/708
1/353 == 1/705 + 1/707
1/359 == 1/561 + 1/997
1/359 == 1/571 + 1/967
1/359 == 1/572 + 1/964
1/359 == 1/576 + 1/953
1/359 == 1/582 + 1/937
1/359 == 1/590 + 1/917
1/359 == 1/599 + 1/896
1/359 == 1/604 + 1/885
1/359 == 1/622 + 1/849
1/359 == 1/635 + 1/826
1/359 == 1/638 + 1/821
1/359 == 1/641 + 1/816
1/359 == 1/646 + 1/808
1/359 == 1/648 + 1/805
1/359 == 1/656 + 1/793
1/359 == 1/658 + 1/790
1/359 == 1/663 + 1/783
1/359 == 1/678 + 1/763
1/359 == 1/682 + 1/758
1/359 == 1/692 + 1/746
1/359 == 1/700 + 1/737
1/359 == 1/714 + 1/722
1/359 == 1/715 + 1/721
1/359 == 1/716 + 1/720
1/359 == 1/717 + 1/719
1/367 == 1/586 + 1/982
1/367 == 1/590 + 1/971
1/367 == 1/593 + 1/963
1/367 == 1/598 + 1/950
1/367 == 1/600 + 1/945
1/367 == 1/605 + 1/933
1/367 == 1/611 + 1/919
1/367 == 1/616 + 1/908
1/367 == 1/625 + 1/889
1/367 == 1/626 + 1/887
1/367 == 1/627 + 1/885
1/367 == 1/628 + 1/883
1/367 == 1/638 + 1/864
1/367 == 1/643 + 1/855
1/367 == 1/647 + 1/848
1/367 == 1/650 + 1/843
1/367 == 1/653 + 1/838
1/367 == 1/656 + 1/833
1/367 == 1/663 + 1/822
1/367 == 1/665 + 1/819
1/367 == 1/667 + 1/816
1/367 == 1/669 + 1/813
1/367 == 1/671 + 1/810
1/367 == 1/681 + 1/796
1/367 == 1/690 + 1/784
1/367 == 1/702 + 1/769
1/367 == 1/708 + 1/762
1/367 == 1/715 + 1/754
1/367 == 1/730 + 1/738
1/367 == 1/731 + 1/737
1/367 == 1/732 + 1/736
1/367 == 1/733 + 1/735
1/373 == 1/600 + 1/986
1/373 == 1/603 + 1/978
1/373 == 1/608 + 1/965
1/373 == 1/610 + 1/960
1/373 == 1/615 + 1/948
1/373 == 1/621 + 1/934
1/373 == 1/626 + 1/923
1/373 == 1/634 + 1/906
1/373 == 1/635 + 1/904
1/373 == 1/636 + 1/902
1/373 == 1/637 + 1/900
1/373 == 1/638 + 1/898
1/373 == 1/639 + 1/896
1/373 == 1/648 + 1/879
1/373 == 1/669 + 1/843
1/373 == 1/714 + 1/781
1/373 == 1/720 + 1/774
1/373 == 1/727 + 1/766
1/373 == 1/742 + 1/750
1/373 == 1/743 + 1/749
1/373 == 1/744 + 1/748
1/373 == 1/745 + 1/747
1/379 == 1/616 + 1/985
1/379 == 1/618 + 1/980
1/379 == 1/620 + 1/975
1/379 == 1/631 + 1/949
1/379 == 1/636 + 1/938
1/379 == 1/644 + 1/921
1/379 == 1/645 + 1/919
1/379 == 1/646 + 1/917
1/379 == 1/647 + 1/915
1/379 == 1/648 + 1/913
1/379 == 1/649 + 1/911
1/379 == 1/650 + 1/909
1/379 == 1/659 + 1/892
1/379 == 1/664 + 1/883
1/379 == 1/668 + 1/876
1/379 == 1/677 + 1/861
1/379 == 1/682 + 1/853
1/379 == 1/684 + 1/850
1/379 == 1/694 + 1/835
1/379 == 1/704 + 1/821
1/379 == 1/710 + 1/813
1/379 == 1/713 + 1/809
1/379 == 1/717 + 1/804
1/379 == 1/721 + 1/799
1/379 == 1/726 + 1/793
1/379 == 1/739 + 1/778
1/379 == 1/754 + 1/762
1/379 == 1/755 + 1/761
1/379 == 1/756 + 1/760
1/379 == 1/757 + 1/759
1/383 == 1/639 + 1/956
1/383 == 1/644 + 1/945
1/383 == 1/664 + 1/905
1/383 == 1/677 + 1/882
1/383 == 1/683 + 1/872
1/383 == 1/688 + 1/864
1/383 == 1/704 + 1/840
1/383 == 1/709 + 1/833
1/383 == 1/721 + 1/817
1/383 == 1/729 + 1/807
1/383 == 1/739 + 1/795
1/383 == 1/747 + 1/786
1/383 == 1/762 + 1/770
1/383 == 1/763 + 1/769
1/383 == 1/764 + 1/768
1/383 == 1/765 + 1/767
1/389 == 1/649 + 1/971
1/389 == 1/654 + 1/960
1/389 == 1/674 + 1/920
1/389 == 1/680 + 1/909
1/389 == 1/684 + 1/902
1/389 == 1/691 + 1/890
1/389 == 1/701 + 1/874
1/389 == 1/711 + 1/859
1/389 == 1/713 + 1/856
1/389 == 1/718 + 1/849
1/389 == 1/723 + 1/842
1/389 == 1/726 + 1/838
1/389 == 1/729 + 1/834
1/389 == 1/745 + 1/814
1/389 == 1/751 + 1/807
1/389 == 1/759 + 1/798
1/389 == 1/774 + 1/782
1/389 == 1/775 + 1/781
1/389 == 1/776 + 1/780
1/389 == 1/777 + 1/779
1/397 == 1/661 + 1/994
1/397 == 1/665 + 1/985
1/397 == 1/666 + 1/983
1/397 == 1/673 + 1/968
1/397 == 1/674 + 1/966
1/397 == 1/675 + 1/964
1/397 == 1/680 + 1/954
1/397 == 1/681 + 1/952
1/397 == 1/682 + 1/950
1/397 == 1/683 + 1/948
1/397 == 1/690 + 1/935
1/397 == 1/691 + 1/933
1/397 == 1/703 + 1/912
1/397 == 1/706 + 1/907
1/397 == 1/711 + 1/899
1/397 == 1/716 + 1/891
1/397 == 1/718 + 1/888
1/397 == 1/720 + 1/885
1/397 == 1/722 + 1/882
1/397 == 1/724 + 1/879
1/397 == 1/726 + 1/876
1/397 == 1/733 + 1/866
1/397 == 1/748 + 1/846
1/397 == 1/752 + 1/841
1/397 == 1/756 + 1/836
1/397 == 1/761 + 1/830
1/397 == 1/767 + 1/823
1/397 == 1/774 + 1/815
1/397 == 1/775 + 1/814
1/397 == 1/789 + 1/799
1/397 == 1/790 + 1/798
1/397 == 1/791 + 1/797
1/397 == 1/792 + 1/796
1/397 == 1/793 + 1/795
1/401 == 1/674 + 1/990
1/401 == 1/695 + 1/948
1/401 == 1/696 + 1/946
1/401 == 1/701 + 1/937
1/401 == 1/705 + 1/930
1/401 == 1/709 + 1/923
1/401 == 1/712 + 1/918
1/401 == 1/720 + 1/905
1/401 == 1/722 + 1/902
1/401 == 1/736 + 1/881
1/401 == 1/741 + 1/874
1/401 == 1/749 + 1/863
1/401 == 1/756 + 1/854
1/401 == 1/764 + 1/844
1/401 == 1/769 + 1/838
1/401 == 1/775 + 1/831
1/401 == 1/782 + 1/823
1/401 == 1/783 + 1/822
1/401 == 1/797 + 1/807
1/401 == 1/798 + 1/806
1/401 == 1/799 + 1/805
1/401 == 1/800 + 1/804
1/401 == 1/801 + 1/803
1/409 == 1/692 + 1/1000
1/409 == 1/693 + 1/998
1/409 == 1/694 + 1/996
1/409 == 1/702 + 1/980
1/409 == 1/703 + 1/978
1/409 == 1/704 + 1/976
1/409 == 1/711 + 1/963
1/409 == 1/712 + 1/961
1/409 == 1/724 + 1/940
1/409 == 1/727 + 1/935
1/409 == 1/737 + 1/919
1/409 == 1/749 + 1/901
1/409 == 1/756 + 1/891
1/409 == 1/759 + 1/887
1/409 == 1/768 + 1/875
1/409 == 1/775 + 1/866
1/409 == 1/790 + 1/848
1/409 == 1/798 + 1/839
1/409 == 1/813 + 1/823
1/409 == 1/814 + 1/822
1/409 == 1/815 + 1/821
1/409 == 1/816 + 1/820
1/409 == 1/817 + 1/819
1/419 == 1/727 + 1/989
1/419 == 1/737 + 1/971
1/419 == 1/740 + 1/966
1/419 == 1/749 + 1/951
1/419 == 1/754 + 1/943
1/419 == 1/756 + 1/940
1/419 == 1/766 + 1/925
1/419 == 1/768 + 1/922
1/419 == 1/773 + 1/915
1/419 == 1/778 + 1/908
1/419 == 1/781 + 1/904
1/419 == 1/784 + 1/900
1/419 == 1/787 + 1/896
1/419 == 1/791 + 1/891
1/419 == 1/799 + 1/881
1/419 == 1/804 + 1/875
1/419 == 1/810 + 1/868
1/419 == 1/818 + 1/859
1/419 == 1/833 + 1/843
1/419 == 1/834 + 1/842
1/419 == 1/835 + 1/841
1/419 == 1/836 + 1/840
1/419 == 1/837 + 1/839
1/421 == 1/732 + 1/991
1/421 == 1/733 + 1/989
1/421 == 1/745 + 1/968
1/421 == 1/748 + 1/963
1/421 == 1/756 + 1/950
1/421 == 1/758 + 1/947
1/421 == 1/772 + 1/926
1/421 == 1/782 + 1/912
1/421 == 1/785 + 1/908
1/421 == 1/788 + 1/904
1/421 == 1/791 + 1/900
1/421 == 1/803 + 1/885
1/421 == 1/808 + 1/879
1/421 == 1/814 + 1/872
1/421 == 1/822 + 1/863
1/421 == 1/837 + 1/847
1/421 == 1/838 + 1/846
1/421 == 1/839 + 1/845
1/421 == 1/840 + 1/844
1/421 == 1/841 + 1/843
1/431 == 1/758 + 1/999
1/431 == 1/761 + 1/994
1/431 == 1/770 + 1/979
1/431 == 1/775 + 1/971
1/431 == 1/791 + 1/947
1/431 == 1/796 + 1/940
1/431 == 1/801 + 1/933
1/431 == 1/804 + 1/929
1/431 == 1/807 + 1/925
1/431 == 1/814 + 1/916
1/431 == 1/818 + 1/911
1/431 == 1/834 + 1/892
1/431 == 1/841 + 1/884
1/431 == 1/842 + 1/883
1/431 == 1/857 + 1/867
1/431 == 1/858 + 1/866
1/431 == 1/859 + 1/865
1/431 == 1/860 + 1/864
1/431 == 1/861 + 1/863
1/433 == 1/766 + 1/996
1/433 == 1/769 + 1/991
1/433 == 1/772 + 1/986
1/433 == 1/777 + 1/978
1/433 == 1/795 + 1/951
1/433 == 1/797 + 1/948
1/433 == 1/805 + 1/937
1/433 == 1/808 + 1/933
1/433 == 1/811 + 1/929
1/433 == 1/818 + 1/920
1/433 == 1/822 + 1/915
1/433 == 1/826 + 1/910
1/433 == 1/831 + 1/904
1/433 == 1/838 + 1/896
1/433 == 1/845 + 1/888
1/433 == 1/846 + 1/887
1/433 == 1/861 + 1/871
1/433 == 1/862 + 1/870
1/433 == 1/863 + 1/869
1/433 == 1/864 + 1/868
1/433 == 1/865 + 1/867
1/439 == 1/785 + 1/996
1/439 == 1/790 + 1/988
1/439 == 1/792 + 1/985
1/439 == 1/802 + 1/970
1/439 == 1/804 + 1/967
1/439 == 1/811 + 1/957
1/439 == 1/814 + 1/953
1/439 == 1/826 + 1/937
1/439 == 1/838 + 1/922
1/439 == 1/843 + 1/916
1/439 == 1/849 + 1/909
1/439 == 1/857 + 1/900
1/439 == 1/858 + 1/899
1/439 == 1/873 + 1/883
1/439 == 1/874 + 1/882
1/439 == 1/875 + 1/881
1/439 == 1/876 + 1/880
1/439 == 1/877 + 1/879
1/443 == 1/796 + 1/999
1/443 == 1/814 + 1/972
1/443 == 1/819 + 1/965
1/443 == 1/824 + 1/958
1/443 == 1/827 + 1/954
1/443 == 1/834 + 1/945
1/443 == 1/846 + 1/930
1/443 == 1/851 + 1/924
1/443 == 1/857 + 1/917
1/443 == 1/865 + 1/908
1/443 == 1/866 + 1/907
1/443 == 1/881 + 1/891
1/443 == 1/882 + 1/890
1/443 == 1/883 + 1/889
1/443 == 1/884 + 1/888
1/443 == 1/885 + 1/887
1/449 == 1/821 + 1/991
1/449 == 1/823 + 1/988
1/449 == 1/828 + 1/981
1/449 == 1/833 + 1/974
1/449 == 1/836 + 1/970
1/449 == 1/839 + 1/966
1/449 == 1/842 + 1/962
1/449 == 1/849 + 1/953
1/449 == 1/853 + 1/948
1/449 == 1/858 + 1/942
1/449 == 1/863 + 1/936
1/449 == 1/869 + 1/929
1/449 == 1/877 + 1/920
1/449 == 1/878 + 1/919
1/449 == 1/892 + 1/904
1/449 == 1/893 + 1/903
1/449 == 1/894 + 1/902
1/449 == 1/895 + 1/901
1/449 == 1/896 + 1/900
1/449 == 1/897 + 1/899
1/457 == 1/843 + 1/998
1/457 == 1/851 + 1/987
1/457 == 1/854 + 1/983
1/457 == 1/861 + 1/974
1/457 == 1/869 + 1/964
1/457 == 1/873 + 1/959
1/457 == 1/878 + 1/953
1/457 == 1/885 + 1/945
1/457 == 1/893 + 1/936
1/457 == 1/894 + 1/935
1/457 == 1/908 + 1/920
1/457 == 1/909 + 1/919
1/457 == 1/910 + 1/918
1/457 == 1/911 + 1/917
1/457 == 1/912 + 1/916
1/457 == 1/913 + 1/915
1/461 == 1/856 + 1/999
1/461 == 1/859 + 1/995
1/461 == 1/862 + 1/991
1/461 == 1/865 + 1/987
1/461 == 1/872 + 1/978
1/461 == 1/881 + 1/967
1/461 == 1/886 + 1/961
1/461 == 1/892 + 1/954
1/461 == 1/893 + 1/953
1/461 == 1/901 + 1/944
1/461 == 1/916 + 1/928
1/461 == 1/917 + 1/927
1/461 == 1/918 + 1/926
1/461 == 1/919 + 1/925
1/461 == 1/920 + 1/924
1/461 == 1/921 + 1/923
1/463 == 1/863 + 1/999
1/463 == 1/866 + 1/995
1/463 == 1/869 + 1/991
1/463 == 1/876 + 1/982
1/463 == 1/880 + 1/977
1/463 == 1/885 + 1/971
1/463 == 1/890 + 1/965
1/463 == 1/896 + 1/958
1/463 == 1/897 + 1/957
1/463 == 1/905 + 1/948
1/463 == 1/920 + 1/932
1/463 == 1/921 + 1/931
1/463 == 1/922 + 1/930
1/463 == 1/923 + 1/929
1/463 == 1/924 + 1/928
1/463 == 1/925 + 1/927
1/467 == 1/877 + 1/999
1/467 == 1/880 + 1/995
1/467 == 1/884 + 1/990
1/467 == 1/888 + 1/985
1/467 == 1/893 + 1/979
1/467 == 1/898 + 1/973
1/467 == 1/904 + 1/966
1/467 == 1/905 + 1/965
1/467 == 1/912 + 1/957
1/467 == 1/913 + 1/956
1/467 == 1/928 + 1/940
1/467 == 1/929 + 1/939
1/467 == 1/930 + 1/938
1/467 == 1/931 + 1/937
1/467 == 1/932 + 1/936
1/467 == 1/933 + 1/935
1/479 == 1/921 + 1/998
1/479 == 1/922 + 1/997
1/479 == 1/928 + 1/990
1/479 == 1/936 + 1/981
1/479 == 1/937 + 1/980
1/479 == 1/952 + 1/964
1/479 == 1/953 + 1/963
1/479 == 1/954 + 1/962
1/479 == 1/955 + 1/961
1/479 == 1/956 + 1/960
1/479 == 1/957 + 1/959
1/487 == 1/952 + 1/997
1/487 == 1/953 + 1/996
1/487 == 1/968 + 1/980
1/487 == 1/969 + 1/979
1/487 == 1/970 + 1/978
1/487 == 1/971 + 1/977
1/487 == 1/972 + 1/976
1/487 == 1/973 + 1/975
1/491 == 1/976 + 1/988
1/491 == 1/977 + 1/987
1/491 == 1/978 + 1/986
1/491 == 1/979 + 1/985
1/491 == 1/980 + 1/984
1/491 == 1/981 + 1/983
1/499 == 1/996 + 1/1000
1/499 == 1/997 + 1/999

Körde som sagt bara ner till 1/1000 för summeringarna.

Senast ändrad: 31 mar 2013 20:41

Dave37 Nivå 13 · Mästerfuskare 6 910 inlägg
medlem sedan 2006

31 mar 2013 #14

Det där kan inte stämma, WA säger att de enda bråken som finns för 1/479 är:

1/480 + 1/229920

1/958 + 1/958

Alltså talföljden

480, 958, 229920

Så det måste vara något avrundningsfel eller nått i ditt program. Felet blir i storleksordningen 10^-8, så se till att ditt program är jävligt noggrant, annars kommer det snabbt misslyckas att räkna så stora bråk.

Senast ändrad: 31 mar 2013 19:05

WASD Nivå 16 · Legendfuskare 11 841 inlägg
medlem sedan 2004

31 mar 2013 #15

Som sagt så räknade jag bara ner till 1/1000. Inte 1/229920. Det skulle ta enormt mycket längre tid.

Varje enskilt bråk tar n^2/2 iterationer med min algoritm. Tar för lång tid om jag ska räkna så högt.

Dave37 Nivå 13 · Mästerfuskare 6 910 inlägg
medlem sedan 2006

31 mar 2013 #16

Då låter vi bli tal lägre än 1/100 då för säkerhets skull. Kan du generera fram en lite längre lista än den jag har i #12 med liknande uppställning men då måste du ha med de som är "halva"?

Senast ändrad: 31 mar 2013 19:32

WASD Nivå 16 · Legendfuskare 11 841 inlägg
medlem sedan 2004

31 mar 2013 #17

Ensamma bråken 1/1 till 1/100 inklusive 1/(2b)+1/(2b) med summeringsförsök ner till 1/10000 i samma format som inlägg #12.

1:
1/1 = 1/2 + 1/2
2,

2:
1/2 = 1/3 + 1/6
1/2 = 1/4 + 1/4
3, 4, 6,

3:
1/3 = 1/4 + 1/12
1/3 = 1/6 + 1/6
4, 6, 12,

4:
1/4 = 1/5 + 1/20
1/4 = 1/6 + 1/12
1/4 = 1/8 + 1/8
5, 6, 8, 12, 20,

5:
1/5 = 1/6 + 1/30
1/5 = 1/10 + 1/10
6, 10, 30,

6:
1/6 = 1/7 + 1/42
1/6 = 1/8 + 1/24
1/6 = 1/9 + 1/18
1/6 = 1/10 + 1/15
1/6 = 1/12 + 1/12
7, 8, 9, 10, 12, 15, 18, 24, 42,

7:
1/7 = 1/8 + 1/56
1/7 = 1/14 + 1/14
8, 14, 56,

8:
1/8 = 1/9 + 1/72
1/8 = 1/10 + 1/40
1/8 = 1/12 + 1/24
1/8 = 1/16 + 1/16
9, 10, 12, 16, 24, 40, 72,

9:
1/9 = 1/10 + 1/90
1/9 = 1/12 + 1/36
1/9 = 1/18 + 1/18
10, 12, 18, 36, 90,

10:
1/10 = 1/11 + 1/110
1/10 = 1/12 + 1/60
1/10 = 1/14 + 1/35
1/10 = 1/15 + 1/30
1/10 = 1/20 + 1/20
11, 12, 14, 15, 20, 30, 35, 60, 110,

11:
1/11 = 1/12 + 1/132
1/11 = 1/22 + 1/22
12, 22, 132,

12:
1/12 = 1/13 + 1/156
1/12 = 1/14 + 1/84
1/12 = 1/15 + 1/60
1/12 = 1/16 + 1/48
1/12 = 1/18 + 1/36
1/12 = 1/20 + 1/30
1/12 = 1/21 + 1/28
1/12 = 1/24 + 1/24
13, 14, 15, 16, 18, 20, 21, 24, 28, 30, 36, 48, 60, 84, 156,

13:
1/13 = 1/14 + 1/182
1/13 = 1/26 + 1/26
14, 26, 182,

14:
1/14 = 1/15 + 1/210
1/14 = 1/16 + 1/112
1/14 = 1/18 + 1/63
1/14 = 1/21 + 1/42
1/14 = 1/28 + 1/28
15, 16, 18, 21, 28, 42, 63, 112, 210,

15:
1/15 = 1/16 + 1/240
1/15 = 1/18 + 1/90
1/15 = 1/20 + 1/60
1/15 = 1/24 + 1/40
1/15 = 1/30 + 1/30
16, 18, 20, 24, 30, 40, 60, 90, 240,

16:
1/16 = 1/17 + 1/272
1/16 = 1/18 + 1/144
1/16 = 1/20 + 1/80
1/16 = 1/24 + 1/48
1/16 = 1/32 + 1/32
17, 18, 20, 24, 32, 48, 80, 144, 272,

17:
1/17 = 1/18 + 1/306
1/17 = 1/34 + 1/34
18, 34, 306,

18:
1/18 = 1/19 + 1/342
1/18 = 1/20 + 1/180
1/18 = 1/21 + 1/126
1/18 = 1/22 + 1/99
1/18 = 1/24 + 1/72
1/18 = 1/27 + 1/54
1/18 = 1/30 + 1/45
1/18 = 1/36 + 1/36
19, 20, 21, 22, 24, 27, 30, 36, 45, 54, 72, 99, 126, 180, 342,

19:
1/19 = 1/20 + 1/380
1/19 = 1/38 + 1/38
20, 38, 380,

20:
1/20 = 1/21 + 1/420
1/20 = 1/22 + 1/220
1/20 = 1/24 + 1/120
1/20 = 1/25 + 1/100
1/20 = 1/28 + 1/70
1/20 = 1/30 + 1/60
1/20 = 1/36 + 1/45
1/20 = 1/40 + 1/40
21, 22, 24, 25, 28, 30, 36, 40, 45, 60, 70, 100, 120, 220, 420,

21:
1/21 = 1/22 + 1/462
1/21 = 1/24 + 1/168
1/21 = 1/28 + 1/84
1/21 = 1/30 + 1/70
1/21 = 1/42 + 1/42
22, 24, 28, 30, 42, 70, 84, 168, 462,

22:
1/22 = 1/23 + 1/506
1/22 = 1/24 + 1/264
1/22 = 1/26 + 1/143
1/22 = 1/33 + 1/66
1/22 = 1/44 + 1/44
23, 24, 26, 33, 44, 66, 143, 264, 506,

23:
1/23 = 1/24 + 1/552
1/23 = 1/46 + 1/46
24, 46, 552,

24:
1/24 = 1/25 + 1/600
1/24 = 1/26 + 1/312
1/24 = 1/27 + 1/216
1/24 = 1/28 + 1/168
1/24 = 1/30 + 1/120
1/24 = 1/32 + 1/96
1/24 = 1/33 + 1/88
1/24 = 1/36 + 1/72
1/24 = 1/40 + 1/60
1/24 = 1/42 + 1/56
1/24 = 1/48 + 1/48
25, 26, 27, 28, 30, 32, 33, 36, 40, 42, 48, 56, 60, 72, 88, 96, 120, 168, 216, 312, 600,

25:
1/25 = 1/26 + 1/650
1/25 = 1/30 + 1/150
1/25 = 1/50 + 1/50
26, 30, 50, 150, 650,

26:
1/26 = 1/27 + 1/702
1/26 = 1/28 + 1/364
1/26 = 1/30 + 1/195
1/26 = 1/39 + 1/78
1/26 = 1/52 + 1/52
27, 28, 30, 39, 52, 78, 195, 364, 702,

27:
1/27 = 1/28 + 1/756
1/27 = 1/30 + 1/270
1/27 = 1/36 + 1/108
1/27 = 1/54 + 1/54
28, 30, 36, 54, 108, 270, 756,

28:
1/28 = 1/29 + 1/812
1/28 = 1/30 + 1/420
1/28 = 1/32 + 1/224
1/28 = 1/35 + 1/140
1/28 = 1/36 + 1/126
1/28 = 1/42 + 1/84
1/28 = 1/44 + 1/77
1/28 = 1/56 + 1/56
29, 30, 32, 35, 36, 42, 44, 56, 77, 84, 126, 140, 224, 420, 812,

29:
1/29 = 1/30 + 1/870
1/29 = 1/58 + 1/58
30, 58, 870,

30:
1/30 = 1/31 + 1/930
1/30 = 1/32 + 1/480
1/30 = 1/33 + 1/330
1/30 = 1/34 + 1/255
1/30 = 1/35 + 1/210
1/30 = 1/36 + 1/180
1/30 = 1/39 + 1/130
1/30 = 1/40 + 1/120
1/30 = 1/42 + 1/105
1/30 = 1/45 + 1/90
1/30 = 1/48 + 1/80
1/30 = 1/50 + 1/75
1/30 = 1/55 + 1/66
1/30 = 1/60 + 1/60
31, 32, 33, 34, 35, 36, 39, 40, 42, 45, 48, 50, 55, 60, 66, 75, 80, 90, 105, 120, 130, 180, 210, 255, 330, 480, 930,

31:
1/31 = 1/32 + 1/992
1/31 = 1/62 + 1/62
32, 62, 992,

32:
1/32 = 1/33 + 1/1056
1/32 = 1/34 + 1/544
1/32 = 1/36 + 1/288
1/32 = 1/40 + 1/160
1/32 = 1/48 + 1/96
1/32 = 1/64 + 1/64
33, 34, 36, 40, 48, 64, 96, 160, 288, 544, 1056,

33:
1/33 = 1/34 + 1/1122
1/33 = 1/36 + 1/396
1/33 = 1/42 + 1/154
1/33 = 1/44 + 1/132
1/33 = 1/66 + 1/66
34, 36, 42, 44, 66, 132, 154, 396, 1122,

34:
1/34 = 1/35 + 1/1190
1/34 = 1/36 + 1/612
1/34 = 1/38 + 1/323
1/34 = 1/51 + 1/102
1/34 = 1/68 + 1/68
35, 36, 38, 51, 68, 102, 323, 612, 1190,

35:
1/35 = 1/36 + 1/1260
1/35 = 1/40 + 1/280
1/35 = 1/42 + 1/210
1/35 = 1/60 + 1/84
1/35 = 1/70 + 1/70
36, 40, 42, 60, 70, 84, 210, 280, 1260,

36:
1/36 = 1/37 + 1/1332
1/36 = 1/38 + 1/684
1/36 = 1/39 + 1/468
1/36 = 1/40 + 1/360
1/36 = 1/42 + 1/252
1/36 = 1/44 + 1/198
1/36 = 1/45 + 1/180
1/36 = 1/48 + 1/144
1/36 = 1/52 + 1/117
1/36 = 1/54 + 1/108
1/36 = 1/60 + 1/90
1/36 = 1/63 + 1/84
1/36 = 1/72 + 1/72
37, 38, 39, 40, 42, 44, 45, 48, 52, 54, 60, 63, 72, 84, 90, 108, 117, 144, 180, 198, 252, 360, 468, 684, 1332,

37:
1/37 = 1/38 + 1/1406
1/37 = 1/74 + 1/74
38, 74, 1406,

38:
1/38 = 1/39 + 1/1482
1/38 = 1/40 + 1/760
1/38 = 1/42 + 1/399
1/38 = 1/57 + 1/114
1/38 = 1/76 + 1/76
39, 40, 42, 57, 76, 114, 399, 760, 1482,

39:
1/39 = 1/40 + 1/1560
1/39 = 1/42 + 1/546
1/39 = 1/48 + 1/208
1/39 = 1/52 + 1/156
1/39 = 1/78 + 1/78
40, 42, 48, 52, 78, 156, 208, 546, 1560,

40:
1/40 = 1/41 + 1/1640
1/40 = 1/42 + 1/840
1/40 = 1/44 + 1/440
1/40 = 1/45 + 1/360
1/40 = 1/48 + 1/240
1/40 = 1/50 + 1/200
1/40 = 1/56 + 1/140
1/40 = 1/60 + 1/120
1/40 = 1/65 + 1/104
1/40 = 1/72 + 1/90
1/40 = 1/80 + 1/80
41, 42, 44, 45, 48, 50, 56, 60, 65, 72, 80, 90, 104, 120, 140, 200, 240, 360, 440, 840, 1640,

41:
1/41 = 1/42 + 1/1722
1/41 = 1/82 + 1/82
42, 82, 1722,

42:
1/42 = 1/43 + 1/1806
1/42 = 1/44 + 1/924
1/42 = 1/45 + 1/630
1/42 = 1/46 + 1/483
1/42 = 1/48 + 1/336
1/42 = 1/49 + 1/294
1/42 = 1/51 + 1/238
1/42 = 1/54 + 1/189
1/42 = 1/56 + 1/168
1/42 = 1/60 + 1/140
1/42 = 1/63 + 1/126
1/42 = 1/70 + 1/105
1/42 = 1/78 + 1/91
1/42 = 1/84 + 1/84
43, 44, 45, 46, 48, 49, 51, 54, 56, 60, 63, 70, 78, 84, 91, 105, 126, 140, 168, 189, 238, 294, 336, 483, 630, 924, 1806,

43:
1/43 = 1/44 + 1/1892
1/43 = 1/86 + 1/86
44, 86, 1892,

44:
1/44 = 1/45 + 1/1980
1/44 = 1/46 + 1/1012
1/44 = 1/48 + 1/528
1/44 = 1/52 + 1/286
1/44 = 1/55 + 1/220
1/44 = 1/60 + 1/165
1/44 = 1/66 + 1/132
1/44 = 1/88 + 1/88
45, 46, 48, 52, 55, 60, 66, 88, 132, 165, 220, 286, 528, 1012, 1980,

45:
1/45 = 1/46 + 1/2070
1/45 = 1/48 + 1/720
1/45 = 1/50 + 1/450
1/45 = 1/54 + 1/270
1/45 = 1/60 + 1/180
1/45 = 1/70 + 1/126
1/45 = 1/72 + 1/120
1/45 = 1/90 + 1/90
46, 48, 50, 54, 60, 70, 72, 90, 120, 126, 180, 270, 450, 720, 2070,

46:
1/46 = 1/47 + 1/2162
1/46 = 1/48 + 1/1104
1/46 = 1/50 + 1/575
1/46 = 1/69 + 1/138
1/46 = 1/92 + 1/92
47, 48, 50, 69, 92, 138, 575, 1104, 2162,

47:
1/47 = 1/48 + 1/2256
1/47 = 1/94 + 1/94
48, 94, 2256,

48:
1/48 = 1/49 + 1/2352
1/48 = 1/50 + 1/1200
1/48 = 1/51 + 1/816
1/48 = 1/52 + 1/624
1/48 = 1/54 + 1/432
1/48 = 1/56 + 1/336
1/48 = 1/57 + 1/304
1/48 = 1/60 + 1/240
1/48 = 1/64 + 1/192
1/48 = 1/66 + 1/176
1/48 = 1/72 + 1/144
1/48 = 1/80 + 1/120
1/48 = 1/84 + 1/112
1/48 = 1/96 + 1/96
49, 50, 51, 52, 54, 56, 57, 60, 64, 66, 72, 80, 84, 96, 112, 120, 144, 176, 192, 240, 304, 336, 432, 624, 816, 1200, 2352,

49:
1/49 = 1/50 + 1/2450
1/49 = 1/56 + 1/392
1/49 = 1/98 + 1/98
50, 56, 98, 392, 2450,

50:
1/50 = 1/51 + 1/2550
1/50 = 1/52 + 1/1300
1/50 = 1/54 + 1/675
1/50 = 1/55 + 1/550
1/50 = 1/60 + 1/300
1/50 = 1/70 + 1/175
1/50 = 1/75 + 1/150
1/50 = 1/100 + 1/100
51, 52, 54, 55, 60, 70, 75, 100, 150, 175, 300, 550, 675, 1300, 2550,

51:
1/51 = 1/52 + 1/2652
1/51 = 1/54 + 1/918
1/51 = 1/60 + 1/340
1/51 = 1/68 + 1/204
1/51 = 1/102 + 1/102
52, 54, 60, 68, 102, 204, 340, 918, 2652,

52:
1/52 = 1/53 + 1/2756
1/52 = 1/54 + 1/1404
1/52 = 1/56 + 1/728
1/52 = 1/60 + 1/390
1/52 = 1/65 + 1/260
1/52 = 1/68 + 1/221
1/52 = 1/78 + 1/156
1/52 = 1/104 + 1/104
53, 54, 56, 60, 65, 68, 78, 104, 156, 221, 260, 390, 728, 1404, 2756,

53:
1/53 = 1/54 + 1/2862
1/53 = 1/106 + 1/106
54, 106, 2862,

54:
1/54 = 1/55 + 1/2970
1/54 = 1/56 + 1/1512
1/54 = 1/57 + 1/1026
1/54 = 1/58 + 1/783
1/54 = 1/60 + 1/540
1/54 = 1/63 + 1/378
1/54 = 1/66 + 1/297
1/54 = 1/72 + 1/216
1/54 = 1/81 + 1/162
1/54 = 1/90 + 1/135
1/54 = 1/108 + 1/108
55, 56, 57, 58, 60, 63, 66, 72, 81, 90, 108, 135, 162, 216, 297, 378, 540, 783, 1026, 1512, 2970,

55:
1/55 = 1/56 + 1/3080
1/55 = 1/60 + 1/660
1/55 = 1/66 + 1/330
1/55 = 1/80 + 1/176
1/55 = 1/110 + 1/110
56, 60, 66, 80, 110, 176, 330, 660, 3080,

56:
1/56 = 1/57 + 1/3192
1/56 = 1/58 + 1/1624
1/56 = 1/60 + 1/840
1/56 = 1/63 + 1/504
1/56 = 1/64 + 1/448
1/56 = 1/70 + 1/280
1/56 = 1/72 + 1/252
1/56 = 1/84 + 1/168
1/56 = 1/88 + 1/154
1/56 = 1/105 + 1/120
1/56 = 1/112 + 1/112
57, 58, 60, 63, 64, 70, 72, 84, 88, 105, 112, 120, 154, 168, 252, 280, 448, 504, 840, 1624, 3192,

57:
1/57 = 1/58 + 1/3306
1/57 = 1/60 + 1/1140
1/57 = 1/66 + 1/418
1/57 = 1/76 + 1/228
1/57 = 1/114 + 1/114
58, 60, 66, 76, 114, 228, 418, 1140, 3306,

58:
1/58 = 1/59 + 1/3422
1/58 = 1/60 + 1/1740
1/58 = 1/62 + 1/899
1/58 = 1/87 + 1/174
1/58 = 1/116 + 1/116
59, 60, 62, 87, 116, 174, 899, 1740, 3422,

59:
1/59 = 1/60 + 1/3540
1/59 = 1/118 + 1/118
60, 118, 3540,

60:
1/60 = 1/61 + 1/3660
1/60 = 1/62 + 1/1860
1/60 = 1/63 + 1/1260
1/60 = 1/64 + 1/960
1/60 = 1/65 + 1/780
1/60 = 1/66 + 1/660
1/60 = 1/68 + 1/510
1/60 = 1/69 + 1/460
1/60 = 1/70 + 1/420
1/60 = 1/72 + 1/360
1/60 = 1/75 + 1/300
1/60 = 1/76 + 1/285
1/60 = 1/78 + 1/260
1/60 = 1/80 + 1/240
1/60 = 1/84 + 1/210
1/60 = 1/85 + 1/204
1/60 = 1/90 + 1/180
1/60 = 1/96 + 1/160
1/60 = 1/100 + 1/150
1/60 = 1/105 + 1/140
1/60 = 1/108 + 1/135
1/60 = 1/110 + 1/132
1/60 = 1/120 + 1/120
61, 62, 63, 64, 65, 66, 68, 69, 70, 72, 75, 76, 78, 80, 84, 85, 90, 96, 100, 105, 108, 110, 120, 132, 135, 140, 150, 160, 180, 204, 210, 240, 260, 285, 300, 360, 420, 460, 510, 660, 780, 960, 1260, 1860, 3660,

61:
1/61 = 1/62 + 1/3782
1/61 = 1/122 + 1/122
62, 122, 3782,

62:
1/62 = 1/63 + 1/3906
1/62 = 1/64 + 1/1984
1/62 = 1/66 + 1/1023
1/62 = 1/93 + 1/186
1/62 = 1/124 + 1/124
63, 64, 66, 93, 124, 186, 1023, 1984, 3906,

63:
1/63 = 1/64 + 1/4032
1/63 = 1/66 + 1/1386
1/63 = 1/70 + 1/630
1/63 = 1/72 + 1/504
1/63 = 1/84 + 1/252
1/63 = 1/90 + 1/210
1/63 = 1/112 + 1/144
1/63 = 1/126 + 1/126
64, 66, 70, 72, 84, 90, 112, 126, 144, 210, 252, 504, 630, 1386, 4032,

64:
1/64 = 1/65 + 1/4160
1/64 = 1/66 + 1/2112
1/64 = 1/68 + 1/1088
1/64 = 1/72 + 1/576
1/64 = 1/80 + 1/320
1/64 = 1/96 + 1/192
1/64 = 1/128 + 1/128
65, 66, 68, 72, 80, 96, 128, 192, 320, 576, 1088, 2112, 4160,

65:
1/65 = 1/66 + 1/4290
1/65 = 1/70 + 1/910
1/65 = 1/78 + 1/390
1/65 = 1/90 + 1/234
1/65 = 1/130 + 1/130
66, 70, 78, 90, 130, 234, 390, 910, 4290,

66:
1/66 = 1/67 + 1/4422
1/66 = 1/68 + 1/2244
1/66 = 1/69 + 1/1518
1/66 = 1/70 + 1/1155
1/66 = 1/72 + 1/792
1/66 = 1/75 + 1/550
1/66 = 1/77 + 1/462
1/66 = 1/78 + 1/429
1/66 = 1/84 + 1/308
1/66 = 1/88 + 1/264
1/66 = 1/99 + 1/198
1/66 = 1/102 + 1/187
1/66 = 1/110 + 1/165
1/66 = 1/132 + 1/132
67, 68, 69, 70, 72, 75, 77, 78, 84, 88, 99, 102, 110, 132, 165, 187, 198, 264, 308, 429, 462, 550, 792, 1155, 1518, 2244, 4422,

67:
1/67 = 1/68 + 1/4556
1/67 = 1/134 + 1/134
68, 134, 4556,

68:
1/68 = 1/69 + 1/4692
1/68 = 1/70 + 1/2380
1/68 = 1/72 + 1/1224
1/68 = 1/76 + 1/646
1/68 = 1/84 + 1/357
1/68 = 1/85 + 1/340
1/68 = 1/102 + 1/204
1/68 = 1/136 + 1/136
69, 70, 72, 76, 84, 85, 102, 136, 204, 340, 357, 646, 1224, 2380, 4692,

69:
1/69 = 1/70 + 1/4830
1/69 = 1/72 + 1/1656
1/69 = 1/78 + 1/598
1/69 = 1/92 + 1/276
1/69 = 1/138 + 1/138
70, 72, 78, 92, 138, 276, 598, 1656, 4830,

70:
1/70 = 1/71 + 1/4970
1/70 = 1/72 + 1/2520
1/70 = 1/74 + 1/1295
1/70 = 1/75 + 1/1050
1/70 = 1/77 + 1/770
1/70 = 1/80 + 1/560
1/70 = 1/84 + 1/420
1/70 = 1/90 + 1/315
1/70 = 1/95 + 1/266
1/70 = 1/98 + 1/245
1/70 = 1/105 + 1/210
1/70 = 1/119 + 1/170
1/70 = 1/120 + 1/168
1/70 = 1/140 + 1/140
71, 72, 74, 75, 77, 80, 84, 90, 95, 98, 105, 119, 120, 140, 168, 170, 210, 245, 266, 315, 420, 560, 770, 1050, 1295, 2520, 4970,

71:
1/71 = 1/72 + 1/5112
1/71 = 1/142 + 1/142
72, 142, 5112,

72:
1/72 = 1/73 + 1/5256
1/72 = 1/74 + 1/2664
1/72 = 1/75 + 1/1800
1/72 = 1/76 + 1/1368
1/72 = 1/78 + 1/936
1/72 = 1/80 + 1/720
1/72 = 1/81 + 1/648
1/72 = 1/84 + 1/504
1/72 = 1/88 + 1/396
1/72 = 1/90 + 1/360
1/72 = 1/96 + 1/288
1/72 = 1/99 + 1/264
1/72 = 1/104 + 1/234
1/72 = 1/108 + 1/216
1/72 = 1/120 + 1/180
1/72 = 1/126 + 1/168
1/72 = 1/136 + 1/153
1/72 = 1/144 + 1/144
73, 74, 75, 76, 78, 80, 81, 84, 88, 90, 96, 99, 104, 108, 120, 126, 136, 144, 153, 168, 180, 216, 234, 264, 288, 360, 396, 504, 648, 720, 936, 1368, 1800, 2664, 5256,

73:
1/73 = 1/74 + 1/5402
1/73 = 1/146 + 1/146
74, 146, 5402,

74:
1/74 = 1/75 + 1/5550
1/74 = 1/76 + 1/2812
1/74 = 1/78 + 1/1443
1/74 = 1/111 + 1/222
1/74 = 1/148 + 1/148
75, 76, 78, 111, 148, 222, 1443, 2812, 5550,

75:
1/75 = 1/76 + 1/5700
1/75 = 1/78 + 1/1950
1/75 = 1/80 + 1/1200
1/75 = 1/84 + 1/700
1/75 = 1/90 + 1/450
1/75 = 1/100 + 1/300
1/75 = 1/120 + 1/200
1/75 = 1/150 + 1/150
76, 78, 80, 84, 90, 100, 120, 150, 200, 300, 450, 700, 1200, 1950, 5700,

76:
1/76 = 1/77 + 1/5852
1/76 = 1/78 + 1/2964
1/76 = 1/80 + 1/1520
1/76 = 1/84 + 1/798
1/76 = 1/92 + 1/437
1/76 = 1/95 + 1/380
1/76 = 1/114 + 1/228
1/76 = 1/152 + 1/152
77, 78, 80, 84, 92, 95, 114, 152, 228, 380, 437, 798, 1520, 2964, 5852,

77:
1/77 = 1/78 + 1/6006
1/77 = 1/84 + 1/924
1/77 = 1/88 + 1/616
1/77 = 1/126 + 1/198
1/77 = 1/154 + 1/154
78, 84, 88, 126, 154, 198, 616, 924, 6006,

78:
1/78 = 1/79 + 1/6162
1/78 = 1/80 + 1/3120
1/78 = 1/81 + 1/2106
1/78 = 1/82 + 1/1599
1/78 = 1/84 + 1/1092
1/78 = 1/87 + 1/754
1/78 = 1/90 + 1/585
1/78 = 1/91 + 1/546
1/78 = 1/96 + 1/416
1/78 = 1/104 + 1/312
1/78 = 1/114 + 1/247
1/78 = 1/117 + 1/234
1/78 = 1/130 + 1/195
1/78 = 1/156 + 1/156
79, 80, 81, 82, 84, 87, 90, 91, 96, 104, 114, 117, 130, 156, 195, 234, 247, 312, 416, 546, 585, 754, 1092, 1599, 2106, 3120, 6162,

79:
1/79 = 1/80 + 1/6320
1/79 = 1/158 + 1/158
80, 158, 6320,

80:
1/80 = 1/81 + 1/6480
1/80 = 1/82 + 1/3280
1/80 = 1/84 + 1/1680
1/80 = 1/85 + 1/1360
1/80 = 1/88 + 1/880
1/80 = 1/90 + 1/720
1/80 = 1/96 + 1/480
1/80 = 1/100 + 1/400
1/80 = 1/105 + 1/336
1/80 = 1/112 + 1/280
1/80 = 1/120 + 1/240
1/80 = 1/130 + 1/208
1/80 = 1/144 + 1/180
1/80 = 1/160 + 1/160
81, 82, 84, 85, 88, 90, 96, 100, 105, 112, 120, 130, 144, 160, 180, 208, 240, 280, 336, 400, 480, 720, 880, 1360, 1680, 3280, 6480,

81:
1/81 = 1/82 + 1/6642
1/81 = 1/84 + 1/2268
1/81 = 1/90 + 1/810
1/81 = 1/108 + 1/324
1/81 = 1/162 + 1/162
82, 84, 90, 108, 162, 324, 810, 2268, 6642,

82:
1/82 = 1/83 + 1/6806
1/82 = 1/84 + 1/3444
1/82 = 1/86 + 1/1763
1/82 = 1/123 + 1/246
1/82 = 1/164 + 1/164
83, 84, 86, 123, 164, 246, 1763, 3444, 6806,

83:
1/83 = 1/84 + 1/6972
1/83 = 1/166 + 1/166
84, 166, 6972,

84:
1/84 = 1/85 + 1/7140
1/84 = 1/86 + 1/3612
1/84 = 1/87 + 1/2436
1/84 = 1/88 + 1/1848
1/84 = 1/90 + 1/1260
1/84 = 1/91 + 1/1092
1/84 = 1/92 + 1/966
1/84 = 1/93 + 1/868
1/84 = 1/96 + 1/672
1/84 = 1/98 + 1/588
1/84 = 1/100 + 1/525
1/84 = 1/102 + 1/476
1/84 = 1/105 + 1/420
1/84 = 1/108 + 1/378
1/84 = 1/112 + 1/336
1/84 = 1/120 + 1/280
1/84 = 1/126 + 1/252
1/84 = 1/132 + 1/231
1/84 = 1/133 + 1/228
1/84 = 1/140 + 1/210
1/84 = 1/147 + 1/196
1/84 = 1/156 + 1/182
1/84 = 1/168 + 1/168
85, 86, 87, 88, 90, 91, 92, 93, 96, 98, 100, 102, 105, 108, 112, 120, 126, 132, 133, 140, 147, 156, 168, 182, 196, 210, 228, 231, 252, 280, 336, 378, 420, 476, 525, 588, 672, 868, 966, 1092, 1260, 1848, 2436, 3612, 7140,

85:
1/85 = 1/86 + 1/7310
1/85 = 1/90 + 1/1530
1/85 = 1/102 + 1/510
1/85 = 1/110 + 1/374
1/85 = 1/170 + 1/170
86, 90, 102, 110, 170, 374, 510, 1530, 7310,

86:
1/86 = 1/87 + 1/7482
1/86 = 1/88 + 1/3784
1/86 = 1/90 + 1/1935
1/86 = 1/129 + 1/258
1/86 = 1/172 + 1/172
87, 88, 90, 129, 172, 258, 1935, 3784, 7482,

87:
1/87 = 1/88 + 1/7656
1/87 = 1/90 + 1/2610
1/87 = 1/96 + 1/928
1/87 = 1/116 + 1/348
1/87 = 1/174 + 1/174
88, 90, 96, 116, 174, 348, 928, 2610, 7656,

88:
1/88 = 1/89 + 1/7832
1/88 = 1/90 + 1/3960
1/88 = 1/92 + 1/2024
1/88 = 1/96 + 1/1056
1/88 = 1/99 + 1/792
1/88 = 1/104 + 1/572
1/88 = 1/110 + 1/440
1/88 = 1/120 + 1/330
1/88 = 1/132 + 1/264
1/88 = 1/152 + 1/209
1/88 = 1/176 + 1/176
89, 90, 92, 96, 99, 104, 110, 120, 132, 152, 176, 209, 264, 330, 440, 572, 792, 1056, 2024, 3960, 7832,

89:
1/89 = 1/90 + 1/8010
1/89 = 1/178 + 1/178
90, 178, 8010,

90:
1/90 = 1/91 + 1/8190
1/90 = 1/92 + 1/4140
1/90 = 1/93 + 1/2790
1/90 = 1/94 + 1/2115
1/90 = 1/95 + 1/1710
1/90 = 1/96 + 1/1440
1/90 = 1/99 + 1/990
1/90 = 1/100 + 1/900
1/90 = 1/102 + 1/765
1/90 = 1/105 + 1/630
1/90 = 1/108 + 1/540
1/90 = 1/110 + 1/495
1/90 = 1/115 + 1/414
1/90 = 1/117 + 1/390
1/90 = 1/120 + 1/360
1/90 = 1/126 + 1/315
1/90 = 1/135 + 1/270
1/90 = 1/140 + 1/252
1/90 = 1/144 + 1/240
1/90 = 1/150 + 1/225
1/90 = 1/165 + 1/198
1/90 = 1/171 + 1/190
1/90 = 1/180 + 1/180
91, 92, 93, 94, 95, 96, 99, 100, 102, 105, 108, 110, 115, 117, 120, 126, 135, 140, 144, 150, 165, 171, 180, 190, 198, 225, 240, 252, 270, 315, 360, 390, 414, 495, 540, 630, 765, 900, 990, 1440, 1710, 2115, 2790, 4140, 8190,

91:
1/91 = 1/92 + 1/8372
1/91 = 1/98 + 1/1274
1/91 = 1/104 + 1/728
1/91 = 1/140 + 1/260
1/91 = 1/182 + 1/182
92, 98, 104, 140, 182, 260, 728, 1274, 8372,

92:
1/92 = 1/93 + 1/8556
1/92 = 1/94 + 1/4324
1/92 = 1/96 + 1/2208
1/92 = 1/100 + 1/1150
1/92 = 1/108 + 1/621
1/92 = 1/115 + 1/460
1/92 = 1/138 + 1/276
1/92 = 1/184 + 1/184
93, 94, 96, 100, 108, 115, 138, 184, 276, 460, 621, 1150, 2208, 4324, 8556,

93:
1/93 = 1/94 + 1/8742
1/93 = 1/96 + 1/2976
1/93 = 1/102 + 1/1054
1/93 = 1/124 + 1/372
1/93 = 1/186 + 1/186
94, 96, 102, 124, 186, 372, 1054, 2976, 8742,

94:
1/94 = 1/95 + 1/8930
1/94 = 1/96 + 1/4512
1/94 = 1/98 + 1/2303
1/94 = 1/141 + 1/282
1/94 = 1/188 + 1/188
95, 96, 98, 141, 188, 282, 2303, 4512, 8930,

95:
1/95 = 1/96 + 1/9120
1/95 = 1/100 + 1/1900
1/95 = 1/114 + 1/570
1/95 = 1/120 + 1/456
1/95 = 1/190 + 1/190
96, 100, 114, 120, 190, 456, 570, 1900, 9120,

96:
1/96 = 1/97 + 1/9312
1/96 = 1/98 + 1/4704
1/96 = 1/99 + 1/3168
1/96 = 1/100 + 1/2400
1/96 = 1/102 + 1/1632
1/96 = 1/104 + 1/1248
1/96 = 1/105 + 1/1120
1/96 = 1/108 + 1/864
1/96 = 1/112 + 1/672
1/96 = 1/114 + 1/608
1/96 = 1/120 + 1/480
1/96 = 1/128 + 1/384
1/96 = 1/132 + 1/352
1/96 = 1/144 + 1/288
1/96 = 1/160 + 1/240
1/96 = 1/168 + 1/224
1/96 = 1/192 + 1/192
97, 98, 99, 100, 102, 104, 105, 108, 112, 114, 120, 128, 132, 144, 160, 168, 192, 224, 240, 288, 352, 384, 480, 608, 672, 864, 1120, 1248, 1632, 2400, 3168, 4704, 9312,

97:
1/97 = 1/98 + 1/9506
1/97 = 1/194 + 1/194
98, 194, 9506,

98:
1/98 = 1/99 + 1/9702
1/98 = 1/100 + 1/4900
1/98 = 1/102 + 1/2499
1/98 = 1/105 + 1/1470
1/98 = 1/112 + 1/784
1/98 = 1/126 + 1/441
1/98 = 1/147 + 1/294
1/98 = 1/196 + 1/196
99, 100, 102, 105, 112, 126, 147, 196, 294, 441, 784, 1470, 2499, 4900, 9702,

99:
1/99 = 1/100 + 1/9900
1/99 = 1/102 + 1/3366
1/99 = 1/108 + 1/1188
1/99 = 1/110 + 1/990
1/99 = 1/126 + 1/462
1/99 = 1/132 + 1/396
1/99 = 1/180 + 1/220
1/99 = 1/198 + 1/198
100, 102, 108, 110, 126, 132, 180, 198, 220, 396, 462, 990, 1188, 3366, 9900,

100:
1/100 = 1/102 + 1/5100
1/100 = 1/104 + 1/2600
1/100 = 1/105 + 1/2100
1/100 = 1/108 + 1/1350
1/100 = 1/110 + 1/1100
1/100 = 1/116 + 1/725
1/100 = 1/120 + 1/600
1/100 = 1/125 + 1/500
1/100 = 1/140 + 1/350
1/100 = 1/150 + 1/300
1/100 = 1/180 + 1/225
1/100 = 1/200 + 1/200
102, 104, 105, 108, 110, 116, 120, 125, 140, 150, 180, 200, 225, 300, 350, 500, 600, 725, 1100, 1350, 2100, 2600, 5100,

BUILD SUCCESSFUL (total time: 1 minute 11 seconds)

Senast ändrad: 31 mar 2013 20:16

Dave37 Nivå 13 · Mästerfuskare 6 910 inlägg
medlem sedan 2006

31 mar 2013 #18

Jag skrev min egna kod (18 rader) och fick fram denna matrisen för de olika nämnarna för bråken 1/1 till 1/20

Motsatta siffror ger bråket i fråga, t.ex. om man tar 12 kolumnen, och så tar man fjärde nämnaren uppifrån (16) och fjärde nedifrån (48) så borde detta resultera i 1/12 = 1/16 + 1/48

Vilket det gör.

Vad jag inte fattar är varför antalet ändras så mycket. Ok om det hade ökat på något trevligt speciellt sätt, men det här ser ju bara ut som värsta röran.

Senast ändrad: 31 mar 2013 20:33

Mezox Nivå 11 · Veteranfuskare 6 465 inlägg
medlem sedan 2004

1 apr 2013 Trådstartare #19

Jag tittade inte här igår kväll så kunde inte vara med i diskussionen men jo jag har skype. :p

Som ni har märkt har 1/p för p primtal bara två sätt att skrivas som en sådan summa. Jag kan avslöja att det finns sätt att förhand (utan räknare eller dator) hitta representationer av 1/n för rimligt stora n (upp till några tusen) inom några minuter.

Dave: I din tabell ser jag ett väldigt intressant mönster, kan man kanske generalisera detta?

Dave37 Nivå 13 · Mästerfuskare 6 910 inlägg
medlem sedan 2006

1 apr 2013 #20

Alltså jag är övertygad om att antalet sätt att skriva det på har med primtalsfaktorer att göra. Eller menar du att första numret alltid är ett högre än den nämnaren man valt? Alltså att 1/n = 1/(n+1) + 1/b? Sista numret följer också (1+n)(2+n), så en av lösningarna är alltid

1/n = 1/(n+1) + 1/((n+1)(2+n))

Senast ändrad: 1 apr 2013 13:31