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andOtherResults<ۍ XQ ff cmr12Joshua/Zelinsky QThe/HopkinsSchodCol \New/Haven,CT06515 dUSA Email:D>߆T ff cmtt12Lord 㠉 ff r/UBern@hotmail.com N cmbx12Abstract."XQ cmr12ApSositivreg cmmi12niscalleda%}h! cmsl12taunumbSerifW(n)!",
cmsy10jn,ރwhere۹isthenumbSer-of- divisors|function.ColtonconjecturedthatthenrumbSer|oftaunrumbers|URnisatleastFuR|{Y cmr81R z @2
"n9(n).InZthispapSerIshorwthatColton'sconjectureistrueforallsucientlylargen.@IalsoprovevXariousotherresultsabSouttaunrumbersandtheirgeneralizations.jhtml: html:'N G cmbx121(Inutro =ductionb#Kennedy,andCoSoper,[html:color push cmyk 0 1 0 03 html: color pop]denedapSositivreintegertobSeataunumbSerifW(n)jn,=whereisthenrumbSer-of-divisorsfunction.8Therstfewtaunrumbersare ~1;2;8;9;12;18;24;36;40;56;60;72;80;:::ʜ;it2isSloane'ssequencehhtml:color push cmyk 0 1 0 0A033950뀉 z ,ю, html: color pop.0cAmongotherthings,IKennedyandCoSoper2shorwedthetau nrumbSershavedensityzero.TheiconceptoftaunrumbSeriwasrediscoveredbyColton,whocalledthesenumbSersrefac-torableO[html:color push cmyk 0 1 0 01 html: color pop].mThispapSerisprimarilyconcernedwithtrwoOconjecturesmadebryColton.Coltonconjectured}thatthenrumbSer}oftaunrumbers}lessthanorequaltoagivrennwasatleasthalfthe+nrumbSerofprimeslessthanorequalton.@InthispaperI+shorwthatColton'sconjectureistrue{forallsucienrtlylargenbyprovingageneralizedversionoftheconjecture.|IfcalculateanuppSerboundforcounrterexamplesof7:4210213 .ColtonalsoconjecturedthattherearenothreeconsecutivretaunumbSersandIshowthistobSethecase.Otherresultsarealsogivren,includingthepropertiesofthetaunrumbersascomparedtotheprimes.8VVariousgeneralizationsofthetaunrumbSersarealsodiscussed. :9color push Black 1G color pop *!T@P:9color push Blackhtml:color push gray 0 color pop html:G color pop:9@Phtml: html: 2(Basiczresults&b#Denitions./Letn9(n)bSethenrumberofprimeslessthanorequalton./LetTƹ(n)bethe nrumbSeroftaunrumberslessthanorequalton. Usingthisnotation,Colton'sconjecturebSecomes:8Tƹ(n)URn9(n)=2foralln.Before
wreproveaslightlyweakerformofthisconjecture,VwementionsomefollowingminorpropSertiesofthetaunrumbers.ThroughoutthispapSer,thefollorwingbasicresult[html:color push cmyk 0 1 0 02 html: color pop,Theorem273]isusedextensively: html: html: color push BlackProp` osition 1. color pop(@ cmti12If35nUR=pn2 cmmi8aqAa cmr61/č1 .vpnaq2/č2.t.pptai?; cmmi6kk
thenW(n)=(a̿1j+1)(a̿2+1)(a̿3+1)(ak:+1). ThenextvretheoremsareallduetoColton.html: html:color push BlackTheorem 2. color popA2ny35offddtaunumberisaperfectsquare. color push BlackPrffoof. color pop$NAssumeDthatnisanoSddtaunrumber.E-LetDn팹=pnaq1/č1 .vpnaq2/č2.t.pptai?kk .ByDPropSositionhtml:color push cmyk 0 1 0 01 html: color popand theP_denitionoftaunrumbSerP_(a̿1^9+51)(a̿2+1)(a̿3+1):::ʜ(akǹ+1)P_jn. jThereforeforanry0