Cycles when iterating x <- phi(sigma(x)), as in A095955. By Judson McCranie, August 2, 2024 Known cycle lengths: 1-9, 11, 12, 15, 16, 18, 19, 21, 22, 23, 29, 31, 53, 56, 80, 93, 167, 351, 595 The first term is the smallest member of the cycle. Cycles whose first term is < 10^12 are known to be all cycles with a member under 10^12. Some of the larger cycles are based on results of Hiroaki Yamanouchi and Charles Greathouse, reported in A095956. Cycles of length 1: See A001229 ------------------- 1 2 8 12 128 240 720 6912 32768 142560 712800 1140480 1190400 3345408 3571200 5702400 14859936 29719872 50319360 118879488 2147483648 3889036800 4389396480 21946982400 47416320000 92177326080 133145026560 331914240000 460886630400 665725132800 Cycles of length 2: See A373435 (lists the first terms) ------------------- 4,6 48,60 72,96 432,480 1728,2016 10368,14080 184320,196560 1658880,1801800 6220800,8363520 10222080,10368000 12856320000,14343436800 3852635996160,4702924800000 4869303828480,5605813204992 Cycles of length 3: See A373453 (lists the first terms) ------------------- 16,30,24 1200,1860,1536 15552,18144,15840 67392,83160,92160 272160,290304,290400 69672960000,105694848000,107909480448 2704853606400,3118300416000,3052010078208 4672651788288000,4747571232768000,4731059935641600 Cycles of length 4: See A373454 (lists the first terms) ------------------- 576,1512,1280,864 41472,72600,51840,42240 2142720000,2935296000,3311642880,3185049600 3233260800,3762339840,3919104000,3450470400 130767436800000,169859767633920,161993115648000,152165744640000 Cycles of length 5: ------------------- 339026688000000,377975808000000,424639621324800,483184764518400,453984583680000 6634509269055173050761216000,7521613519844726223667200000,7946886558074859593662464000,7794495412499746337587200000,7970172471593905204651622400 Cycles of length 6: ------------------- 1800,2880,3024,3840,3456,2560 27648,30976,54432,48384,55296,34560 Cycles of length 7: ------------------- 1357815495178340352000,1580133611691048960000,1588475981255933952000,1480028644284825600000, 1444230382133837168640,1451126554305232896000,1400452293678700953600 Cycles of length 8: ------------------- 12959170560000,13694917017600,15199382661120,13352544092160,14087779200000,14112536739840,15801827328000,15164916940800 140005324800000,154781093376000,154581073920000,181803061440000,173877580800000,215694943027200,199832371200000,164339004211200 Cycles of length 9: -------------------- 113218560,124895232,163296000,181149696,170698752,125798400,116121600,139708800,136857600 Cycles of length 11: -------------------- 326592,550368,435456,580608,851840,552960,524160,442368,432000,368640,381024 Cycles of length 12: -------------------- 5033164800,6808043520,6291456000,6220800000,5761082880,5225472000,6657251328,5283532800,5837832000, 7608287232,7429968000,6521389056 Cycles of length 15: -------------------- 40255488,48384000,43130880,41912640,47029248,70253568,91998720,82944000,83825280,71663616,52428800, 79221120,70778880,57600000,42456960 Cycles of length 16: -------------------- 15906579537576591360,18045342419779584000,19325000115412992000,25726733926143344640,23892362919936000000, 20661064807219200000,19075633660506931200,19704349609426944000,21044157285924864000,24301630247731200000, 20907461836800000000,18384933079115366400,16352995065200640000,21515155200000000000,17833942322380800000, 18516979468999065600 Cycles of length 18: -------------------- 150493593600,152374763520,202491394560,167215104000,219847799808,161864220672,247328774784,191102976000, 207622711296,178362777600,283740364800,233003796480,221908377600,204838502400,214695936000,237283098624, 185794560000,178886400000 52575215616000,65953585152000,72425041920000,67421129932800,105646502707200,146114230026240,147938964848640, 165974054400000,162627139584000,154067816448000,129491376537600,109559336140800,121950602403840, 97844914814976,62620384296960,63779291136000,60340406353920,56804297932800 Cycles of length 19: -------------------- 5912762368345595046789120000,7254832008896939009507328000,7279466748540788268859392000, 7213126974530390065152000000,10194904551406779745422336000,10089265696756103997554688000, 8997578388379919702021898240,8424469010015531827200000000,8016005617861116395520000000, 8922025831350888467728957440,9321681806063537856970752000,8294552237301669122015232000, 7817689349095548245350809600,7340582740338551309598720000,6766690961670222643200000000, 8506421072997623878975488000,7773892255206604800000000000,6503347138025138813927424000, 6489336428789760000000000000 Cycles of length 21: -------------------- 12227604480,14265538560,12541132800,12582912000,12878611200,14108774400,25945920000,32199552000,34256977920,27378819072,26824089600,29628426240,23710579200,20570112000,22145587200,18341406720,15328051200,13589544960,14756742000,18475776000,15895885824 Cycles of length 22: -------------------- 1316603904000,1433272320000,1593188352000,1937738880000,2360519884800,2711072851200,2738983403520,3039876532224,2225424015360,2799886279680,2955021189120,3485665094400,2844328919040,2348237537280,2222131968000,1907506298880,2087078400000,2211158476800,2006581248000,1747955220480,1513281945600,1348093393920 Cycles of length 23: -------------------- 4706367188052108902400000,4737788853587693980876800,6167945373624796446720000,7401571774423940277964800, 7451577453027262464000000,6815004995467103816908800,6689987470357954560000000,5666677173780480000000000, 6790357033533618564096000,5818882309331528083046400,8114633258189570703360000,7691996109325530562560000, 7107097549855732531200000,7163660823112765440000000,7260939933731724931891200,7856604933218347725619200, 9529887508593172807680000,7054087416761548800000000,6081842394698770022400000,5617432647196996534272000, 5060202702848354628403200,4960651419823901560012800,5135898283721879678484480 Cycles of length 29: -------------------- 4491065730939531087052800000,5843473415374159414886400000,4805805774834298257408000000, 5387697732406839607296000000,6818329333667412836352000000,8482739623602733685145600000, 7633514077645626880819200000,7707435464456247273062400000,8253159020458444121702400000, 9136531976619706634207232000,8901163370975858031329280000,11538276008362966646784000000, 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1148431566029745827296051200000,1380887284953678092793481789440,1257616018907620954387513344000, 1307882312881495519851970560000,1093171371345363979468800000000,1113364795757273579520000000000, 1556006600664907776000000000000,1589087117716315139173515264000,1521648744500628933731942400000, 1473610001979555478241280000000,1530319520831660101730304000000,1455845701839592957673472000000, 1249849501610035798396108800000,1364595378655365070612070400000,1013581285375443122257920000000, 953181555019042447687680000000,830401468496187631652044800000,787288108983242973926916096000, 994178400680970089083699200000,992844841580199164604579840000,932364388989389131776000000000, 1141471655628948452105256960000,969813718221921347174400000000,1101968184456510366140006400000, 1139066012971656806400000000000,1062619630816483835117568000000,1125994418187420201438412800000, 1051993434508318996766392320000,1090205576261593989120000000000,939100854598603697915795865600, 1110595551930721626476249088000,969493118645649637638144000000,1071194452020316476341936455680, 1158669013567489966080000000000,1149864451094635930705379328000,1190239244121889862823444480000, 1046140630409436870731209113600,947991020498810044416000000000 Cycles of length 56: -------------------- 423134057722616480079544320000,1105543137258436224693945600000,1192364895898107095128473600000, 1142799176201136148316160000000,1303927187181770420382597120000,1156126047527113850880000000000, 1015186152687177319833600000000,878902332539271118848000000000,793631005222184560676044800000, 1023862899374093830037962752000,919675455821718475379834880000,791229735501820644556800000000, 662655400232209137008640000000,707532731538684688465920000000,868602999461215955852250316800, 895715635908554998677504000000,822269408953970919997440000000,981611852759877567545081856000, 1002198964116612791992320000000,953574640656404181480898560000,922590369316405562572800000000, 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460746791781336386371584000000,465470213672763780680908800000,433538224970155306883481600000, 430086383405786481828456038400,499795255910258605817856000000 Cycles of length 80: -------------------- 552904100512727040000000000000,696503498700308586090725376000,708371305700006760567275520000, 656838978627467162419200000000,698377664437474595545939968000,765385828027288219350466560000, 665339899887208460955156480000,724900790533969837587824640000,709264460069770560012288000000, 622786255725429446934528000000,939646730445332152320000000000,1022803407134360384569344000000, 1505417220240824185395609600000,1245989869690455934894080000000,1133921241418052256399360000000, 988759028538410613008089743360,909569132131369920705331200000,885574719065989467183513600000, 751314290075758679269205606400,1037548404837346372721049600000,923094363817438189977600000000, 921348346747186210406400000000,894236008711252554547200000000,1283452099399778487478832332800, 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