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PZartitionF=unctionsandDivisorSums.s獒 -XQ ff cmr12Neville/Robbins 9 Mathematics/Department bSan/F4ranciscoStateUniversity ZSan/F4rancisco,CA94132 ,html:color push cmyk 0 1 0 0߆T ff cmtt12robbins@math.sfsu.edu html: color pop.s獒 N ff cmbx12Abstract @ cmti12Lffet"qg cmmi12n;rubenaturalnumbers,Xwithr!",
cmsy10URXQ cmr122.~Wepresentconvolution-typeformulasforthenumber ofepffartitionsofnthatare(1)notdivisiblebyrS;|(2)coprimetorS.zA2notherresultobtainedis35aformulaforthesumoftheoffdddivisorsofn.html: html:'N G cmbx121(Inutro =ductionb#WVederivreseveralconvolution-typSeidentitieslinkingpartitionfunctionstodivisorsums,therebryG4extendingsomepriorresults.NInaddition,^WweobtainaLambSertseries-likeidentityforsumsofoSdddivisors. html: html: 2(PreliminariesLetAURN@,thesetofallnaturalnrumbSers..7Letn;m;r2URN+withrUR2;m2;msquarefree.8LetxUR2C5;jxj<1.*(N cmbx12Denition 1뀉 z EǎT5Letp2 cmmi8A(n)denotethenrumbSerofpartitionsofninrtopartsthatbelongtoA.Denition 2뀉 z EǎT5LetA(n)denotethesumofthedivisors,d,ofnsucrhthatdUR2A.Denition 3뀉 z EǎT5Letp(n)denotethenrumbSerofpartitionsofn. :9color push Black 1G color pop *KE&:9color push Blackhtml:color push gray 0 color pop html:G color pop3ڍ&:9Denition 4:9뀉 z Eǎ[p*Letqn9(n)denotethenrumbSerofpartitionsofninrtodistinctparts(orinto :9oSddparts).*:9Denition 5:9뀉 z Eǎ[p*Letq|{Y cmr80(n)denotethenrumbSerofpartitionsofninrtodistinctoddparts(the:9nrumbSerofself-conjugatepartitionsofn).:9Denition 6:9뀉 z Eǎ[p*LetKYbrb(n)denotethenrumbSerKYofr-rffegularpartitionsKYofn(thenrumberKYof:9partitions~ofnsucrhthatnopartisamultipleofrorsuchthatnopartoSccursrormore:9times).9:9Remark::9뀉 z 2n:HNotethatb2(n)UR=qn9(n).:9Denition 7:9뀉 z Eǎ[p*LetfmĹ(n)denotethenrumbSerofpartitionsofnsucrhthatallpartsare:9coprimetom.:9Denition 8:9뀉 z Eǎ[p*Let9rb(n)denotethesumofthedivisors,]Qd,ofnsucrhthatddoSesnotdivide:9rS.:9Denition 9:9뀉 z Eǎ[p*Let2n9K cmsy8RAmĹ(n)denotethesumofthedivisors,d,ofnsucrhthatdiscoprimeto:9m.:9Denition 10:9뀉 z Lǎb0*Let(n)denoteEuler'stotienrtfunction.9:9Remark::9뀉 z 2n:HIfpisprime,thenfp](n)UR=bp(n)and2n9RAp.=(n)UR=p(n). k html: