Odd Multiperfect Numbers

Let n be a positive integer,σ(n):=∑d▕n d.Ifσ(n)=2n,then n is ealled a perfect number.Supposeκ≤2 is an integer.We say n is multiperfect ifσ(n)=kn.In this thesis we study the strueture,Euler's part and distribution of multiperfect numbers.1.Structure of multiperfect numbersLe n be oddκ-perfect,v,2(κ)≥1 be the highest powers such 2 dividingκ,and 1≤s≤v2(κ).Then n=p1e1p2e2…psesM2,where M∈N.pi Primes,(pi,M)=1 and ej are odd.If v2(κ)-s=a1+a2+…+as+b1+b2+…+6s.ai≥0. bj≥0,then P1,…ps satisfy pi≡2ai+1-1(mod 2ai+2).e1,…,es satisfy ej≡2bj+1-1(mod 2bj+2).Ⅱ=p1e1…pses is called the Euler's part of n.The well known Euler's theorem on odd perfect number is an immediately corollary.An odd perfect number n must be n=παm2, whereπis primr,αis odd and(π,m)=1.Moreover:π≡α≡1(mod 4).These results generalize Broughan and Q.Zhou's results(J.Number Theory 128: 1566-1575,2008)2.Properties of Euler's partof odd multiperfect numbersWe obtain congruence properties of the Euler's part of odd perfect numbber n=παM2:σ(M2)≡1(mod 4)<=>π≡α(mod 8),σ(M2)≡3(mod 4)<=>π≡α+4(mod 8).Let n=ПM2 be odd 2k -perfect,Пbe the Euler's part of n.Ifp▕M =>p≡3(mod 4),Π=p1e1…psesq1f1…q2tf2t satisfy(σ(Π),p1…Ps)=1,where pi≡1(mod 4).qj≡3 mod 4 and t≥0,thenΩ(2k/σ(Π))≡(mod 2),whereΩ(σ(п2/2k)counts all prime factors ofσ(П)/2k.Let n=Пq2βПis=1pi2βi be odd 2k-perfect.If q is a Fermat prime,that is q=22t+1 for some integer t≥1,andПis=1(2β+1)≠0(mod q):then q2β▕σ(Π).We extend Starni's results(J.Number Theory 116:483-486,2006；J.Number Theory 37:366-369,1991)。3.The distribution of odd multiperfect numbersWe prove that for any integer r≤1,the number of oddκ-perfeet numbers n withω(n)≤r is bounded by(κ-1)4r3. This generalize Pollack's results on odd perfect numbers(Amer.Math.Monthly.118:161-164,2011)。...