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Anaivesearchforabuheileadyoutoguessthattheabundantnumbersaresimplythemultiplesof6.ly，aerthan6oftheform6nisabundaorsof6nmustiogetherwithn，2n，and3n，whiorethantheihisobservatioeoshowthatabujustaboutsixesaswearguethesameerfeumberk.Thefakwilliherwithallthefactorsoftheperfeumberk，eachmultipliedbynsothatthesumofalltheproperfaktoatleast1+nk，andthereforeanymultipleofaperfeumberwillbeabundant.Forexample，28isperfece2×28=56，3×28=84etc.areallabundant.
Amultiplesofperfeumbersahesametoken，multiplesofabundahemselvesabundant.Havihisdisightguessthatallabundantnumbersaresimplymultiplesofperfeumbers.However，youdoolooktoomuchfurthertofiextothisjecture，for70isabundantbutsfactorsareperfedeed，70isthefirstso-calledweirdexactlyforthisreason（thesourceofthislabelisexplainedbelow）.
&hesediscoveries，youmightstillthi，justasthereseemtobeherearenooddabundaher.Inotherwords，ourmodifiedjecturemightbethatalloddnumbersaredefit.Calofthealiquotsumsofthefirstfewhundredoddnumberswouldseemtothistheory，buttheclaimiseventuallydebuesting945，whichhas97sasthesumofitsproperdivisors.esopenasanymultipleofanabundantnumberisabundant，andinparticulartheoddmultiplesof945immediatelysupplyuswithinfinitelymanymoreoddabundantnumbers.
&alittlemoreshreediscoverthister-examplemorequiifweunthioneodderanother.Foraohavealargealiquotsum，itsoffadlargefactorsatthat，whichthemselvesbeihsmallfactors.Wethereforebuildhlargealiquotsumsbymultiplyingsmallprimestogether.Ifwearefobersonly，weshouldlookatthosethatareproductsofthefirstfewoddprimes，whichare3，5，7，etc.Thisruleofthumbwouldsoootest33×5×7=945andtherebydiscovertheabuyamongtheoddnumbersalso.
Itisnotthatunusualtofindthatthesmallestexampleofahpropertiesturnsouttoberatherlarge.Thisisespeciallytrueifthespecifiedpropertiesimplicitlybuildafactorstrutotherequiredhesmallestexampleturnouttobegigantic，althoughnotnecessarilyhardtofihegiveiesihesolution.Anexampleofanumberriddleofthiskindistofiisfivetimesadthreetimesafifthpower.Theansweris
7，119，140，625=5×ll253=3×755
Thereasosolutionisinthebilliohardtosee.Anysolutionnhastohavetheform3r5smforsomepositivepowersrandsaheremaiorsarecollectedtogetherintoasihatisnotdivisibleby30r5.Ifwefirstfothepossiblevaluesofr，weobservethatsiimesacube，theexpobeamultipleof3，aimesa5thpower，thenumberr-1hastobeamultipleof 5.Thesmallestrthatsatisfiesboththeseultaneouslyisr=6.Iheexposhastobeamultipleof5，whiles--1hastobeamutipleof3ahatfitsthebilliss=10.Tomakenassmallaspossible，wetakem=1andson=36×510=3（3×52）5=3×755，sothatimesa5thpoweraimen=5（32×53）3=5×11253，andsonisalso5timesacube.
Aneveremeexampleisthecelebrated，attributedtoArchimedes（287–212BC），thegreatestmathematitiquity.Itwashe19.Thesmallestherdofcattlethatsatisfiesalltheimposedtsintheinal44-linepoemisrepresentedbyahover200，000digits!
Awarningtobegleanedfromallthisisthatdisplaytheirfullvarietyuotherealmse.Forthatreasohattherearehfewerthan300digitsdoesnotinitselfgivegrthatthey‘probably’do.Allthesame，itisthecasethatsomeleadihefieldwouldbeastonishedifournedup.
&urothegeneralbehaviourofaliquotsequeillsimplequestionsthatmaybeputthatnoonesossibilitiesareopentoaliquotsequehesequesaprime，itwillimmediatelytermi1，andotdothisinanyotherway.Ifthisdoeshesequencecouldbedsorepresentasoumber.Thereis，however，aedpossibilitythatisrevealedbygthealiquotsequenceof95:
95=5×19→（1+5+19）=25=5×5→（1+5）=6→6→6→….
ealthough95isnotitselfasoumber，itsaliquotsequeuallyhitsasoumber（ormoreprethiscase，theperfeumber6）andtheoacycle.
Thereisceivablyonepossibilityremaining，thatbeingthatthealiquotsequenbersaprimenorasoumber，inwhichcasethesequebeanunendingseriesofdifferentnumbers，noneofwhichareeitherprimeorsociable.Isthispossible?Surprisingly，nooneknows.Whatismisthattherearesmallnumberswhosealiquotsequenunknown（andtherebyremaindidatesfsufisequeofthesemysteriousnumbersis276，whosesequens:
276→396→696→1104→1872→3770→3790→
→3050→2716→2772→…
butnoolywhereitendsup.
Itmightwellbethatthereaderwouldliketoexplorealittleontheirown，inwhichcaseIshouldletyouiofhowtocalculatetheso-calledaliquotfun）fromtheprimefactorizationofofallterms（pk+1-1）（p-1），wherepkisthehighestprimepoweroftheprimepthatdividesraitself.Forexample，276=22×3×23andso
asiheseisequencefor276listedabove.
Thereishetypesofwetroducebygivihebeararelationshiptothealiquotfun.Aswehavealreadymentioifa（n）=nandabundantifa（numberisthesumofsomeofitsproperdivisors（thoselessthann），soitfollowsfromthedenitionthatallsemiperfeumbersareeitherperfedant.Forexample，18issemiperfectas18=3+6+9.Anumberiscalledweirdifitisabundantbut，aweirdnumberis70.
&akethevieiingtoomiseousiowiherarbitrarilydenedbersdoesnotofitsownaakethemiing.Weshouldkop.Thatsaid，itisregthattheurategiesusedtotackletheseioremiofwhatEudEulershowedusioperfeumbers.YouwillrecallthatwhatEuclidprovedwasthatifaMersennenumberrimethenanothernumbererfect.Eulerthenprovedverselythatallevenumbersarisefromthisapproathe9thtury，thePersiaiThabitibnQurraintrodubernatripleofnumberswhich，ifallprime，allowedthestruicablepair.Thabit’sstruwasgeherbyEulerihtury，buteventhisenhanulatiooyieldafeairsandtherearemanyamicablepairsthatdohisstru.（Therearenownearly12millionknownpairsofamiumbers.）Iimes，asimilarapprivesastruofweirdnumbersfromumbersshouldtheyhappentobeprime，andthisformulahassuccessfullyfeweirdhmorethanftydigits.
&tershaveservedtofamiliarizethereaderwithfadfactorizatiouralnumbers，orpositiveiheyarealsoknown，illustratedthroughavarietyofexamples.Thiswillstandyouiheupingchapter，inwhichyouwilllearnhowthoseideasareappliedtopraphy，thesceofsecrets.
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