on some sequences of integersnew

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1、MATHEMATICSOFCOMPUTATIONVolume65,Number214April1996,Pages837{840d-COMPLETESEQUENCESOFINTEGERSP.ERDOSANDMORDECHAILEWIN}Abstract.Aninnitesequencea1

2、γnonnegative.1.IntroductionAninnitesequenceofintegersa1

3、itiveintegersiscomplete.Cassels[2]considerablygeneralizestheresultofBirch.Inthispaperweareconcernedwithd-completenessofsetsoftheformfpqgandfpqrγgwith;;γnonnegative.ThiswasmotivatedbyaquestionaskedbyPaulErd}os:Isittruethateveryinteger>1isthesumofdistinctintegersoftheform23(andnonnegativeintegers)w

4、herenosummanddividestheother?"Overestimatingthedicultyoftheproblem,hetoldittoJansenandwroteittoLewin.Jansenalmostimmediatelygaveasimpleproofbyinduction,whichwasalsofoundbyLewinandbyseveralotherstowhomErd}oswroteortoldtheproblem.Forthesakeofcompletenessweshallreproducethesimpleproofinthecurrentpaper

5、.Seealso[3].2.ThemainresultsProposition1(AppearedalsoasaQuickie"in[3]).Thesequencef23gisd-complete.Proof.Upton=3thepropositionclearlyholds.Nowletalltheintegersupton,3p

6、andsomisrepresentableand3pdoesnotdivideanysummandrepresentingm.Thennisrepresentable.Thisprovestheproposition.ReceivedbytheeditorJanuary30,1994and,inrevisedform,August3,1994,February12,1995,andMarch16,1995.1991MathematicsSubjectClassication.Primary11B13.c1996AmericanMathematicalSociety837838P.ERDOSA

7、NDMORDECHAILEWIN}Thequestionwhetherfornlargeenoughitcanalwaysbewrittenintheformn=a+a+


+awhereallthea'sareoftheform23andallareinan12kinterval(x;2x)yieldsanegativeanswer,sincethenumberofintegersofthef