the abundancy index of divisors of spoof odd perfect numbers

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   `   r    V    c   v   1   s   u    b   m    c    t    /    0    2    9    9    3    4    5    W   m   `    t    f  .    O    ^    ]    <    M   `   r    2    9    0    5 ^FE @BUON@ODS CONEV KJ NCZCTKRT KJ TYKKJ KNN YERJED^ OUMBERT Akse @ro`hnk B. Nrcs 0 Lr`nu`te kj M. Td. M`tfem`tcds, Ne H` T`hhe Uocverscty, M`och`, Yfchcppcoes akse`ro`hnkbnrcsIlm`ch.dkm Re decv en1 , Revcse n1 , @d depten1 , Yubhcsfen1 @bstr`dt \e d`hh  o  ` spkkj knn perjedt oumber cj  o  cs knn `on  o  =  gm  jkr twk cotelers g,m ? 0 sudf tf`t  τ (g)(m + 0) = 2o, wfere  τ  cs tfe sum-kj -ncvcskrs juodtcko. Co tfcs p`per, we sfkw fkw resuhts `o`hklkus tk tfkse kj knn perjedt oumbers dkuhn be est`bhcsfen jkr spkkj knn perjedt oumbers (ktferwcse gokwo co tfe hcter`ture `s Nesd`r tes oumbers). Co p`rtcduh`r, we prencdt tf`t `o `o`hklu e kj tfe Nesd`rtes- J reocd he-Tkrhc dkoaedture jkr knn perjed t oum bers fkhns jkr spkkj knn perjed t oum- bers. J urtfermkre, we dkoaed ture tf`t tfe qu`sc-Euher prcme kj ` spkkj k nn perjedt oum ber cs `hsk cts h`rlest qu`sc- prcme j`dtkr. 0. Cot rknudtcko Cj  O  cs ` pksctcve coteler, tfeo we wrcte  τ(O ) jkr tfe sum kj tfe ncvcskrs kj  O . @ oumber O  c s perjedt  cj  τ(O ) = 2O . @ oumber M  cs  `hmkst perjedt  cj  τ(M ) = 2M 0. Ct cs durreothy uogokwo wfetfer tfere `re cooctehy m`oy eveo perjedt oumbers, kr wfetf er `oy knn p erjed t oumber s (KYOs) excst. Ko tfe ktfer f`on, ct cs gokwo tf`t pkwers kj twk (codhuncol 0) `re `hmkst perjedt, `htfkulf ct cs stchh `o kpeo prkbhem tk ruhe kut ktfer pksscbhe jkrms kj eveo `hmkst perjedt oumbers, `on `hsk, cj 0 wchh be tfe kohy knn `hmkst perjedt oumber tk be ‘ncsdkveren”. Kdfem `on R`k redeothy prkven  W09 ] tf`t, cj  O  cs `o knn perjedt oumber, tfeo O ? 09 0599 `on tf`t tfe h`rlest dkmpkoeot (c.e., ncvcskr  p ` wctf  p  prcme) kj  O  cs bcller tf`o 09 :2 . ^fcs cmprkves ko prevckus resuhts by Breot, Dkfeo `on te Rcehe W2]  co 0330 (O ? 09 499 ) `on Dkfeo  W4]  co 036< (h`rlest dkmpkoeot  p ` ? 09 29 ). @o knn perjedt oumber  O  =  q r t 2 cs s`cn tk be lcveo co Euherc`o jkrm cj  q  cs prcme wctf q  ≠ r 0 (mkn >) `on ldn( q, t) = 0. (^fe oumber  q  cs d`hhen tfe Euher prcme , wfche tfe dkmpkoeot  q r cs rejerren tk `s tfe  Euher j`dtkr . Okte tf`t, scode  q cs prcme `on  q  ≠ 0 (mkn >), tfeo  q  ≥ 5.) 0 Lhkb`h Dhceot Tuppkrt Tpedc`hcst, @nveot Tkjtw`re - ^eheperjkrm`ode Yfchcppcoes

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Page 1: The Abundancy Index of Divisors of Spoof Odd Perfect Numbers

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