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Perfetti speculari (numeri)

Rappresentazione dei numeri 

Si chiamano “perfetti speculari” i numeri naturali che scritti a rovescio sono uguali alla somma dei loro divisori, escluso il numero stesso, scritti a rovescio. Una definizione equivalente è che se si scrive il numero, il segno uguale e la somma dei divisori, escluso il numero stesso, l’uguaglianza è verificata leggendo da destra a sinistra.

Per esempio, 10311 è perfetto speculare, perché i suoi divisori, escludendo 10311, sono: 1, 3, 7, 21, 491, 1473 e 3437 e 1 + 3 + 7 + 12 + 194 + 3741 + 7343 = 11301.

 

I primi numeri perfetti speculari in base 10 sono: 6, 10311, 21661371, 1460501511, 7980062073, 79862699373, 798006269373 (Joseph L. Pe, Daniel Dockery e Jens Kruse Andersen, The Online Encyclopedia of Integer Sequences http://oeis.org).

 

Jens Kruse Andersen dimostrò che se un numero n formato da 140 seguito da uno o più blocchi della forma (0)m10(9)k89 è primo (detto “primo di Andersen”), 57n è perfetto speculare. Le ripetizioni di 0 e 9 nei vari blocchi non devono necessariamente avere la stessa lunghezza.

Per esempio, 140010999890109989 è primo e 57 • 140010999890109989 = 7980626993736269373 è perfetto speculare.

Con ogni probabilità i primi di Andersen sono infiniti e quindi i numeri perfetti speculari in base 10 sono infiniti.

 

In base 10 non si conosce alcun numero perfetto speculare pari maggiore di 6.

 

Mark Ganson avanzò la congettura che tutti i numeri perfetti speculari in base 10 siano multipli di 3.

 

La definizione si può estendere ad altre basi; la tabella seguente mostra i numeri perfetti speculari fino a 109 nelle basi da 2 a 20 (M. Fiorentini, 2017).

Base

Numeri perfetti speculari

2

102 = 2, 111010001010000011012 = 952845

3

103 = 3, 1013 = 10, 1100113 = 328, 202112023 = 4988, 2021011100023 = 401276, 22220121020223 = 1578590, 100021121122013 = 1644292, 11001101120010213 = 19378312, 11001212012110213 = 19461256, 22012202021000223 = 39282416, 110012120010020213 = 58380568, 1100111120122110213 = 174592456, 1100121201110020213 = 175148872, 11001100112012110213 = 523031128, 11001111201110020213 = 523774168, 21012110002010100023 = 930035684

4

1324 = 30, 2014 = 33, 2314 = 45, 2021014 = 2193, 12133024 = 6642, 113213023114 = 1547445, 102300102322114 = 78662565, 132000002101114 = 125831445,

5

105 = 5, 115 = 6, 2045 = 54, 113415 = 846, 204225 = 1362, 212145 = 1434, 1301415 = 5046, 2144345 = 7494, 21200145 = 35634, 21422345 = 37194, 123421435 = 121548, 204222425 = 170322, 212242145 = 179934, 213402425 = 183822, 213410345 = 183894, 213422225 = 184062, 214202345 = 185694, 214214225 = 185862, 214420145 = 187134, 214444345 = 187494, 2040000225 = 843762, 2040022225 = 844062, 2041022345 = 847194, 2042224425 = 851622, 2120000145 = 890634, 2120140225 = 891762, 2120234225 = 892362, 2121400225 = 896262, 2121410345 = 896394, 2122200145 = 898134, 2122224345 = 898494, 2122420145 = 899634, 2134002425 = 918822, 2134242225 = 920562, 2134244425 = 920622, 2142002345 = 928194, 2142014225 = 928362, 2142140225 = 929262, 2143400225 = 933762, 2144340225 = 936762, 2144434225 = 937362, 20400000225 = 4218762, 20400104345 = 4219494, 20401022345 = 4222194, 20402210345 = 4226394, 20402444425 = 4228122, 20410202345 = 4235694, 20410222145 = 4235934, 20422230345 = 4257894, 20424200225 = 4263762, 20424422425 = 4265322, 21200002345 = 4453194, 21200220145 = 4454634, 21200244345 = 4454994, 21201402425 = 4458822, 21201422225 = 4459062, 21202420145 = 4462134, 21214010345 = 4481394, 21214102145 = 4481934, 21222024345 = 4490994, 21222142425 = 4491822, 21224420145 = 4499634, 21340010345 = 4593894, 21340104345 = 4594494, 21340230345 = 4595394, 21342200225 = 4601262, 21342224425 = 4601622, 21342304345 = 4601994, 21421400225 = 4646262, 21421422225 = 4646562, 21434024425 = 4669122, 21442002345 = 4678194, 21442024345 = 4678494, 21442142425 = 4679322, 21442220145 = 4679634, 21443410345 = 4683894, 21444340225 = 4686762, 21444422345 = 4687194, 101133434115 = 10293606, 204001022345 = 21097194, 204002224425 = 21101622, 204002442225 = 21103062, 204010422345 = 21112194, 204022022225 = 21131562, 204023020145 = 21134634, 204024224425 = 21139122, 204024442225 = 21140562, 204102222345 = 21179694, 204103422225 = 21184062, 204104200145 = 21185634, 204104214225 = 21185862, 204104224345 = 21185994, 204220000225 = 21281262, 204220242225 = 21283062, 204221044345 = 21284994, 204222222225 = 21289062, 204230340225 = 21299262, 204242242225 = 21320562, 204243022345 = 21322194, 204244210345 = 21326394, 204244224425 = 21326622, 204244444425 = 21328122, 212000222345 = 22267194, 212002202345 = 22273194, 212002224345 = 22273494, 212014002425 = 22293822, 212014024425 = 22294122, 212014102145 = 22294434, 212022022145 = 22303434, 212022140225 = 22304262, 212022222345 = 22304694, 212022244345 = 22304994, 212024200145 = 22310634, 212024222145 = 22310934, 212024442145 = 22312434, 212140010345 = 22406394, 212141044345 = 22409994, 212142210345 = 22413894, 212142422425 = 22415322, 212220234225 = 22454862, 212221410345 = 22458894, 212222202345 = 22460694, 212222214225 = 22460862, 212222222145 = 22460934, 212222422345 = 22462194, 212222444345 = 22462494, 212234022225 = 22481562, 212242024345 = 22490994, 212242222345 = 22492194, 212242242145 = 22492434, 212243400225 = 22496262, 212243410345 = 22496394, 212244202345 = 22498194, 212244420145 = 22499634, 213400102145 = 22969434, 213400244425 = 22970622, 213401044345 = 22972494, 213402302145 = 22976934, 213402304345 = 22976994, 213402430345 = 22977894, 213410214225 = 22985862, 213410422345 = 22987194, 213422000225 = 23006262, 213422024425 = 23006622, 213422104345 = 23006994, 213424420225 = 23015262, 213424442225 = 23015562, 214200022145 = 23203434, 214201422225 = 23209062, 214201424425 = 23209122, 214202214225 = 23210862, 214202422345 = 23212194, 214224340225 = 23249262, 214224420145 = 23249634, 214340242225 = 23345562, 214420142425 = 23391822, 214422202345 = 23398194, 214422342425 = 23399322, 214422434225 = 23399862, 214422442145 = 23399934, 214434000225 = 23418762, 214434024425 = 23419122, 214434244425 = 23420622, 214442002345 = 23428194, 214442014225 = 23428362, 214442140225 = 23429262, 214443422225 = 23434062, 214443424425 = 23434122, 214444224345 = 23435994, 214444342425 = 23436822, 432032020025 = 45756502, 2040001022345 = 105472194, 2040002224425 = 105476622, 2040002302145 = 105476934, 2040002444425 = 105478122, 2040010422345 = 105487194, 2040022230345 = 105507894, 2040022244425 = 105508122, 2040023020145 = 105509634, 2040024304345 = 105514494, 2040024442225 = 105515562, 2040102014225 = 105553362, 2040102220145 = 105554634, 2040103400225 = 105558762, 2040104222145 = 105560934, 2040104342425 = 105561822, 2040104422345 = 105562194, 2040104434225 = 105562362, 2040220024425 = 105656622, 2040220244425 = 105658122, 2040230222145 = 105673434, 2040242024425 = 105694122, 2040243034225 = 105697362, 2040244202425 = 105701322, 2040244210345 = 105701394, 2041020022145 = 105890934, 2041020140225 = 105891762, 2041020244345 = 105892494, 2041021400225 = 105896262, 2041022342425 = 105899322, 2041022422345 = 105899694, 2041034104345 = 105919494, 2041034222425 = 105920322, 2041043402425 = 105933822, 2041044340225 = 105936762, 2041044420145 = 105937134, 2042200104345 = 106406994, 2042200244425 = 106408122, 2042202304345 = 106414494, 2042202430345 = 106415394, 2042210224345 = 106423494, 2042210342425 = 106424322, 2042222000225 = 106443762, 2042222024425 = 106444122, 2042223020145 = 106447134, 2042223042145 = 106447434, 2042223044345 = 106447494, 2042224202425 = 106451322, 2042224430345 = 106452894, 2042302234225 = 106492362, 2042302242145 = 106492434, 2042302244345 = 106492494, 2042303402425 = 106496322, 2042304214225 = 106498362, 2042304420145 = 106499634, 2042304444345 = 106499994, 2042420022225 = 106594062, 2042421034225 = 106597362, 2042421042145 = 106597434, 2042422442225 = 106603062, 2042430444345 = 106612494, 2042442102145 = 106631934, 2042442222425 = 106632822, 2042442230345 = 106632894, 2120000002345 = 111328194, 2120000244345 = 111329994, 2120002420145 = 111337134, 2120002444345 = 111337494, 2120014220225 = 111357762, 2120022234225 = 111367362, 2120024200145 = 111373134, 2120024444345 = 111374994, 2120140002425 = 111468822, 2120140010345 = 111468894, 2120140022225 = 111469062, 2120140102145 = 111469434, 2120140220225 = 111470262, 2120140222425 = 111470322, 2120142304345 = 111476994, 2120142422425 = 111477822, 2120142442225 = 111478062, 2120220014225 = 111515862, 2120220220145 = 111517134, 2120222200145 = 111523134, 2120234230345 = 111545394, 2120234242225 = 111545562, 2120242242145 = 111554934, 2121400220225 = 112032762, 2121400244425 = 112033122, 2121401044345 = 112034994, 2121402222225 = 112039062, 2121402422425 = 112040322, 2121410202345 = 112048194, 2121410224345 = 112048494, 2121410420145 = 112049634, 2121422002425 = 112068822, 2121422102145 = 112069434, 2121424304345 = 112076994, 2121424420225 = 112077762, 2121424430345 = 112077894, 2122200142425 = 112266822, 2122200234225 = 112267362, 2122200242145 = 112267434, 2122202342425 = 112274322, 2122202442145 = 112274934, 2122214002425 = 112293822, 2122214010345 = 112293894, 2122214024425 = 112294122, 2122214220225 = 112295262, 2122214242225 = 112295562, 2122222002345 = 112303194, 2122223400225 = 112308762, 2122223422225 = 112309062, 2122224222145 = 112310934, 2122224434225 = 112312362, 2122341034225 = 112409862, 2122342202425 = 112413822, 2122342222225 = 112414062, 2122420024345 = 112453494, 2122421410345 = 112458894, 2122422202345 = 112460694, 2122422214225 = 112460862, 2122434002425 = 112481322, 2122434102145 = 112481934, 2122434244425 = 112483122, 2122442140225 = 112491762, 2122442222345 = 112492194, 2122442234225 = 112492362, 2122443400225 = 112496262, 2134000102145 = 114844434, 2134000222425 = 114845322, 2134000244425 = 114845622, 2134002200225 = 114851262, 2134002304345 = 114851994, 2134010224345 = 114860994, 2134010342425 = 114861822, 2134010444345 = 114862494, 2134022002425 = 114881322, 2134022220225 = 114882762, 2134023020145 = 114884634, 2134023042145 = 114884934, 2134024222225 = 114889062, 2134102234225 = 114929862, 2134104214225 = 114935862, 2134104222145 = 114935934, 2134220022225 = 115031562, 2134220230345 = 115032894, 2134220242225 = 115033062, 2134221022345 = 115034694, 2134222302145 = 115039434, 2134230222145 = 115048434, 2134230340225 = 115049262, 2134230444345 = 115049994, 2134242010345 = 115068894, 2134244222225 = 115076562, 2134244430345 = 115077894, 2142000220145 = 116017134, 2142000222345 = 116017194, 2142002222145 = 116023434, 2142014222425 = 116045322, 2142022000145 = 116053134, 2142022002345 = 116053194, 2142022024345 = 116053494, 2142022142425 = 116054322, 2142022242145 = 116054934, 2142024214225 = 116060862, 2142140024425 = 116156622, 2142140242225 = 116158062, 2142142420225 = 116165262, 2142142442225 = 116165562, 2142220000145 = 116203134, 2142220022145 = 116203434, 2142220234225 = 116204862, 2142221402425 = 116208822, 2142221424425 = 116209122, 2142222342425 = 116211822, 2142234010345 = 116231394, 2142242014225 = 116240862, 2142244200145 = 116248134, 2142244214225 = 116248362, 2142244224345 = 116248494, 2143400022225 = 116719062, 2143400222425 = 116720322, 2143401034225 = 116722362, 2143402210345 = 116726394, 2143410214225 = 116735862, 2143422002425 = 116756322, 2143422010345 = 116756394, 2143424200225 = 116763762, 2143424430345 = 116765394, 2144200002345 = 116953194, 2144214010345 = 116981394, 2144214102145 = 116981934, 2144214104345 = 116981994, 2144222014225 = 116990862, 2144222024345 = 116990994, 2144222142425 = 116991822, 2144223402425 = 116996322, 2144223410345 = 116996394, 2144224224345 = 116998494, 2144340010345 = 117093894, 2144340222425 = 117095322, 2144342222225 = 117101562, 2144342304345 = 117101994, 2144342422425 = 117102822, 2144342442225 = 117103062, 2144420022145 = 117140934, 2144420140225 = 117141762, 2144420220145 = 117142134, 2144421400225 = 117146262, 2144421424425 = 117146622, 2144422214225 = 117148362, 2144422224345 = 117148494, 2144422342425 = 117149322, 2144422422345 = 117149694, 2144442024345 = 117178494, 2144442234225 = 117179862, 2144442244345 = 117179994, 2144443410345 = 117183894, 2144444422345 = 117187194, 20400000104345 = 527344494, 20400000242225 = 527345562, 20400001034225 = 527347362, 20400002210345 = 527351394, 20400010222145 = 527360934, 20400022022225 = 527381562, 20400022220225 = 527382762, 20400022222425 = 527382822, 20400022230345 = 527382894, 20400023022345 = 527384694, 20400024200225 = 527388762, 20400024430345 = 527390394, 20400102014225 = 527428362, 20400103424425 = 527434122, 20400104224345 = 527435994, 20400104422345 = 527437194, 20400104442145 = 527437434, 20400220220225 = 527532762, 20400222210345 = 527538894, 20400222420225 = 527540262, 20400222430345 = 527540394, 20400230200145 = 527548134, 20400230340225 = 527549262, 20400243022345 = 527572194, 20400243042145 = 527572434, 20400244200225 = 527576262, 20400244444425 = 527578122, 20401020022145 = 527765934, 20401020222345 = 527767194, 20401022342425 = 527774322, 20401022422345 = 527774694, 20401034000225 = 527793762, 20401034022225 = 527794062, 20401034104345 = 527794494, 20401042000145 = 527803134, 20401042002345 = 527803194, 20401042242145 = 527804934, 20401042244345 = 527804994, 20401044202345 = 527810694, 20401044214225 = 527810862, 20401044340225 = 527811762, 20402200222425 = 528282822, 20402201044345 = 528284994, 20402202224425 = 528289122, 20402202430345 = 528290394, 20402210434225 = 528299862, 20402222000225 = 528318762, 20402222024425 = 528319122, 20402223042145 = 528322434, 20402224210345 = 528326394, 20402224302145 = 528326934, 20402304420145 = 528374634, 20402420024425 = 528469122, 20402420222425 = 528470322, 20402420230345 = 528470394, 20402430444345 = 528487494, 20402442002425 = 528506322, 20402442222425 = 528507822, 20402443044345 = 528509994, 20402444442225 = 528515562, 20410200022145 = 529453434, 20410200222345 = 529454694, 20410202340225 = 529461762, 20410202442145 = 529462434, 20410214000225 = 529481262, 20410214022225 = 529481562, 20410222002345 = 529490694, 20410222024345 = 529490994, 20410222142425 = 529491822, 20410224340225 = 529499262, 20410340022225 = 529594062, 20410340230345 = 529595394, 20410341042145 = 529597434, 20410342202425 = 529601322, 20410342442225 = 529603062, 20410420000145 = 529640634, 20410420022145 = 529640934, 20410420142425 = 529641822, 20410420222345 = 529642194, 20410420234225 = 529642362, 20410421400225 = 529646262, 20410422202345 = 529648194, 20410422224345 = 529648494, 20410434002425 = 529668822, 20410434024425 = 529669122, 20410443400225 = 529683762, 20410444222145 = 529685934, 20422000010345 = 532031394, 20422001020145 = 532034634, 20422010200145 = 532048134, 20422010340225 = 532049262, 20422022230345 = 532070394, 20422023034225 = 532072362, 20422024202425 = 532076322, 20422024444425 = 532078122, 20422104434225 = 532124862, 20422220102145 = 532219434, 20422230224345 = 532235994, 20422230434225 = 532237362, 20422242010345 = 532256394, 20422244202425 = 532263822, 20422244304345 = 532264494, 20422244422425 = 532265322, 20423020142425 = 532454322, 20423022200145 = 532460634, 20423022444345 = 532462494, 20423034010345 = 532481394, 20423034242225 = 532483062, 20423034244425 = 532483122, 20423044342425 = 532499322, 20423044434225 = 532499862, 20424202442225 = 532978062, 20424202444425 = 532978122, 20424210202345 = 532985694, 20424210214225 = 532985862, 20424222002425 = 533006322, 20424222010345 = 533006394, 20424222102145 = 533006934, 20424223020145 = 533009634, 20424224430345 = 533015394, 20424302002345 = 533053194, 20424302022145 = 533053434, 20424303400225 = 533058762, 20424304214225 = 533060862, 20424420002425 = 533156322, 20424420102145 = 533156934, 20424420242225 = 533158062, 20424422222225 = 533164062, 20424422304345 = 533164494, 20424430342425 = 533174322, 20424430442145 = 533174934, 20424442024425 = 533194122, 20424442230345 = 533195394, 20424442242225 = 533195562, 21200000222345 = 556642194, 21200001400225 = 556646262, 21200002342425 = 556649322, 21200014024425 = 556669122, 21200014102145 = 556669434, 21200022002345 = 556678194, 21200022222345 = 556679694, 21200024214225 = 556685862, 21200140000225 = 556781262, 21200140024425 = 556781622, 21200140102145 = 556781934, 21200140220225 = 556782762, 21200142202425 = 556788822, 21200142302145 = 556789434, 21200220220145 = 556829634, 21200221410345 = 556833894, 21200222422345 = 556837194, 21200234010345 = 556856394, 21200234022225 = 556856562, 21200234220225 = 556857762, 21200234230345 = 556857894, 21200242022145 = 556865934, 21200242140225 = 556866762, 21200242234225 = 556867362, 21200244342425 = 556874322, 21201400000225 = 557343762, 21201400024425 = 557344122, 21201400244425 = 557345622, 21201401020145 = 557347134, 21201402420225 = 557352762, 21201410224345 = 557360994, 21201410434225 = 557362362, 21201410442145 = 557362434, 21201422022225 = 557381562, 21201422242225 = 557383062, 21201423042145 = 557384934, 21201424202425 = 557388822, 21201424224425 = 557389122, 21201424444425 = 557390622, 21202200140225 = 557579262, 21202201400225 = 557583762, 21202202222145 = 557585934, 21202202442145 = 557587434, 21202214024425 = 557606622, 21202214222425 = 557607822, 21202214230345 = 557607894, 21202223402425 = 557621322, 21202223422225 = 557621562, 21202224222145 = 557623434, 21202340022225 = 557719062, 21202341034225 = 557722362, 21202342420225 = 557727762, 21202420024345 = 557765994, 21202420222345 = 557767194, 21202421400225 = 557771262, 21202421424425 = 557771622, 21202422200145 = 557773134, 21202422224345 = 557773494, 21202422420145 = 557774634, 21202434000225 = 557793762, 21202434024425 = 557794122, 21202434102145 = 557794434, 21202434242225 = 557795562, 21202434244425 = 557795622, 21202444200145 = 557810634, 21202444340225 = 557811762, 21202444442145 = 557812434, 21214000242225 = 560158062, 21214001042145 = 560159934, 21214002202425 = 560163822, 21214002222225 = 560164062, 21214002420225 = 560165262, 21214010340225 = 560174262, 21214010422345 = 560174694, 21214022024425 = 560194122, 21214022104345 = 560194494, 21214023022345 = 560197194, 21214024200225 = 560201262, 21214024202425 = 560201322, 21214024210345 = 560201394, 21214024444425 = 560203122, 21214102014225 = 560240862, 21214102140225 = 560241762, 21214104200145 = 560248134, 21214104442145 = 560249934, 21214220222425 = 560345322, 21214220244425 = 560345622, 21214222200225 = 560351262, 21214230444345 = 560362494, 21214242002425 = 560381322, 21214242220225 = 560382762, 21214243034225 = 560384862, 21222000024345 = 561328494, 21222000234225 = 561329862, 21222000244345 = 561329994, 21222002202345 = 561335694, 21222002222145 = 561335934, 21222014010345 = 561356394, 21222022142425 = 561366822, 21222022222345 = 561367194, 21222022234225 = 561367362, 21222023410345 = 561371394, 21222024202345 = 561373194, 21222024342425 = 561374322, 21222140220225 = 561470262, 21222140230345 = 561470394, 21222142442225 = 561478062, 21222220140225 = 561516762, 21222220244345 = 561517494, 21222221424425 = 561521622, 21222222222145 = 561523434, 21222222434225 = 561524862, 21222222442145 = 561524934, 21222234242225 = 561545562, 21222242000145 = 561553134, 21222242002345 = 561553194, 21223400010345 = 562031394, 21223400022225 = 562031562, 21223400220225 = 562032762, 21223402200225 = 562038762, 21223402422425 = 562040322, 21223402430345 = 562040394, 21223410342425 = 562049322, 21223410420145 = 562049634, 21223410442145 = 562049934, 21223424202425 = 562076322, 21223424422425 = 562077822, 21223424444425 = 562078122, 21224200024345 = 562265994, 21224214002425 = 562293822, 21224214010345 = 562293894, 21224214242225 = 562295562, 21224222002345 = 562303194, 21224223424425 = 562309122, 21224224200145 = 562310634, 21224224214225 = 562310862, 21224340024425 = 562406622, 21224340102145 = 562406934, 21224342430345 = 562415394, 21224342444425 = 562415622, 21224420024345 = 562453494, 21224420234225 = 562454862, 21224420242145 = 562454934, 21224420244345 = 562454994, 21224421402425 = 562458822, 21224422214225 = 562460862, 21224434022225 = 562481562, 21224434220225 = 562482762, 21224442000145 = 562490634, 21224443400225 = 562496262, 21224444202345 = 562498194, 21224444434225 = 562499862, 21340000024425 = 574219122, 21340000102145 = 574219434, 21340001042145 = 574222434, 21340002302145 = 574226934, 21340002430345 = 574227894, 21340010214225 = 574235862, 21340010222145 = 574235934, 21340010434225 = 574237362, 21340022222425 = 574257822, 21340102000145 = 574303134, 21340102234225 = 574304862, 21340103402425 = 574308822, 21340103424425 = 574309122, 21340104442145 = 574312434, 21340220010345 = 574406394, 21340220022225 = 574406562, 21340220102145 = 574406934, 21340220244425 = 574408122, 21340221022345 = 574409694, 21340222304345 = 574414494, 21340230200145 = 574423134, 21340242002425 = 574443822, 21340242244425 = 574445622, 21340243034225 = 574447362, 21340244222225 = 574451562, 21340244304345 = 574451994, 21341020242145 = 574642434, 21341021402425 = 574646322, 21341022202345 = 574648194, 21341022434225 = 574649862, 21341022442145 = 574649934, 21341022444345 = 574649994, 21341034010345 = 574668894, 21341034242225 = 574670562, 21341042222345 = 574679694, 21341043422225 = 574684062, 21341044342425 = 574686822, 21341044422345 = 574687194, 21341044442145 = 574687434, 21342200104345 = 575156994, 21342201022345 = 575159694, 21342210444345 = 575174994, 21342222102145 = 575194434, 21342222230345 = 575195394, 21342224224425 = 575201622, 21342224304345 = 575201994, 21342304342425 = 575249322, 21342304422345 = 575249694, 21342420002425 = 575343822, 21342420010345 = 575343894, 21342420024425 = 575344122, 21342420220225 = 575345262, 21342420244425 = 575345622, 21342421020145 = 575347134, 21342421042145 = 575347434, 21342442024425 = 575381622, 21342443022345 = 575384694, 21342443042145 = 575384934, 21342444442225 = 575390562, 21342444444425 = 575390622, 21420002200145 = 580085634, 21420002214225 = 580085862, 21420002422345 = 580087194, 21420014000225 = 580106262, 21420022024345 = 580115994, 21420022142425 = 580116822, 21420140104345 = 580219494, 21420140242225 = 580220562, 21420220140225 = 580266762, 21420222202345 = 580273194, 21420222224345 = 580273494, 21420234000225 = 580293762, 21420234010345 = 580293894, 21420234102145 = 580294434, 21420242014225 = 580303362, 21420242140225 = 580304262, 21420243422225 = 580309062, 21420243424425 = 580309122, 21420244224345 = 580310994, 21421400000225 = 580781262, 21421400230345 = 580782894, 21421400242225 = 580783062, 21421410340225 = 580799262, 21421422104345 = 580819494, 21421423022345 = 580822194, 21421424224425 = 580826622, 21421424422425 = 580827822, 21422200234225 = 581017362, 21422202214225 = 581023362, 21422214022225 = 581044062, 21422214230345 = 581045394, 21422214244425 = 581045622, 21422222014225 = 581053362, 21422222140225 = 581054262, 21422223400225 = 581058762, 21422224442145 = 581062434, 21422340230345 = 581157894, 21422340244425 = 581158122, 21422342222225 = 581164062, 21422342422425 = 581165322, 21422342442225 = 581165562, 21422422214225 = 581210862, 21422422222145 = 581210934, 21422434000225 = 581231262, 21422434022225 = 581231562, 21422442002345 = 581240694, 21422442024345 = 581240994, 21422442234225 = 581242362, 21422442242145 = 581242434, 21422443402425 = 581246322, 21422444202345 = 581248194, 21422444214225 = 581248362, 21434000022225 = 583594062, 21434001022345 = 583597194, 21434002420225 = 583602762, 21434002444425 = 583603122, 21434010444345 = 583612494, 21434022222425 = 583632822, 21434023022345 = 583634694, 21434023044345 = 583634994, 21434024222225 = 583639062, 21434024422425 = 583640322, 21434024430345 = 583640394, 21434102022145 = 583678434, 21434104214225 = 583685862, 21434104224345 = 583685994, 21434104340225 = 583686762, 21434104342425 = 583686822, 21434104422345 = 583687194, 21434104434225 = 583687362, 21434221020145 = 583784634, 21434222222225 = 583789062, 21434222422425 = 583790322, 21434222444425 = 583790622, 21434230200145 = 583798134, 21434230422345 = 583799694, 21434230434225 = 583799862, 21434242024425 = 583819122, 21434242222425 = 583820322, 21434242230345 = 583820394, 21434243034225 = 583822362, 21434244444425 = 583828122, 21442000022145 = 584765934, 21442000024345 = 584765994, 21442000142425 = 584766822, 21442000242145 = 584767434, 21442001410345 = 584771394, 21442002202345 = 584773194, 21442002224345 = 584773494, 21442002444345 = 584774994, 21442014244425 = 584795622, 21442022002345 = 584803194, 21442022244345 = 584804994, 21442023424425 = 584809122, 21442140000225 = 584906262, 21442140102145 = 584906934, 21442141020145 = 584909634, 21442141034225 = 584909862, 21442142210345 = 584913894, 21442220000145 = 584953134, 21442222222145 = 584960934, 21442222444345 = 584962494, 21442234010345 = 584981394, 21442234230345 = 584982894, 21442242000145 = 584990634, 21442242014225 = 584990862, 21442243400225 = 584996262, 21442243402425 = 584996322, 21442243410345 = 584996394, 21442244420145 = 584999634, 21442244442145 = 584999934, 21442244444345 = 584999994, 21443400010345 = 585468894, 21443401034225 = 585472362, 21443402202425 = 585476322, 21443402420225 = 585477762, 21443410224345 = 585485994, 21443410342425 = 585486822, 21443410422345 = 585487194, 21443410434225 = 585487362, 21443422104345 = 585506994, 21443423042145 = 585509934, 21443424202425 = 585513822, 21443424442225 = 585515562, 21444200002345 = 585703194, 21444202200145 = 585710634, 21444202422345 = 585712194, 21444214000225 = 585731262, 21444214104345 = 585731994, 21444214222425 = 585732822, 21444222000145 = 585740634, 21444222242145 = 585742434, 21444223410345 = 585746394, 21444224202345 = 585748194, 21444340230345 = 585845394, 21444341022345 = 585847194, 21444342210345 = 585851394, 21444420014225 = 585890862, 21444420022145 = 585890934, 21444420142425 = 585891822, 21444444340225 = 585936762, 21444444434225 = 585937362

6

446 = 28, 4016 = 145, 41016 = 901, 50016 = 1081, 1013003116 = 1749715, 231224320016 = 153533665, 234422141436 = 159114735

7

67 = 6, 107 = 7, 217 = 15, 4366446367 = 26347866, 16222211347 = 76863630

8

68 = 6

9

69 = 6, 1839 = 156, 2649 = 220, 2727469 = 166092, 135608419 = 6712588, 2714741469 = 120390828, 3371120349 = 147276256,

10

6, 10311, 21661371

11

611 = 6, 1011 = 11, 13211 = 156

12

612 = 6, A0112 = 1441, 3266612 = 66606, 199550412 = 5421460

13

613 = 6, 1013 = 13, 2213 = 28, 2413 = 30, 33B7113 = 94225, 2745A213 = 952278, 1577831413 = 89699640

14

614 = 6, C0114 = 2353

15

615 = 6, 6115 = 91, 4E29615 = 250341

16

616 = 6, 10A116 = 4257, 1E052816 = 1967400, 2DE15C116 = 48108993, C135ACB16 = 202595019

17

617 = 6, 1017 = 17

18

618 = 6, 42118 = 1333

19

619 = 6, 1019 = 19

20

620 = 6, B120 = 221, I0120 = 7201, I6I18620 = 58704566

 

Curioso, e al momento inspiegato, il fatto che in base 5 vi siano molti più numeri perfetti speculari che in tutte le altre basi fino a 20 insieme, almeno fino a 109.

 

Tutti i numeri perfetti sono perfetti speculari nelle basi maggiori del numero stesso; per esempio, 6 è perfetto speculare in tutte le basi maggiori di 6.

 

Ogni numero primo p è perfetto speculare in base p.

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