Sigma,USigma,UPhi 2

[franktaw at netscape.net]

You lost the k0 in there.  There are no solutions to sigma(m)^2 = 
uSigma(m) * uPhi(m)

It would be better to state this in terms of divisibility; you are 
looking for uSigma(m) * uPhi(m) divides sigma(m)^2, etc.

Franklin T. Adams-Watters


-----Original Message-----
From: zbi74583 at boat.zero.ad.jp

      Hi,Seqfans
    I considered about the equations as follows.

        f_i(m)*f_j(m)=k0*f_k(m)*f_l(m) , for some integer k0
        where 0<=i,j,k,l<=3
              f_0(m)=Sigma(m)
              f_1(m)=UnitarySigma(m)
              f_2(m)=UnitaryPhi(m)
              f_3(m)=m

     Ordinary, these equations have a few solutions, but the following 
three
equations have exceptionally many solutions.

         Sigma(m)^2= UnitarySigma(m)*UnitaryPhi(m)     .... E.1
         UnitarySigma(m)^2= Sigma(m)*UnitaryPhi(m)     .... E.2
         Sigma(m)*UnitarySigma(m)= UnitaryPhi(m)^2     .... E.3

    The solutions  are as  follows.

    S_1 :
         
1,2,3,6,14,15,30,35,42,70,78,105,190,210,348*,357,418,570,714,910,1045,12
54,2090,2730,....

    S_2 :
         
1,2,3,6,14,15,30,35,42,70,78,105,190,210,357,418,570,714,910,1045,1254,19
76*,2090,2730,....

    S_3 :
         
1,2,3,6,14,15,30,35,42,70,78,105,190,210,312*,357,418,570,714,910,1045,12
54,1428*,2090,....


    These sequences are similar, but different. See the terms with "*".

     The reason why they have the same terms  is that if a solution is 
square
free then Sigma(m) and UnitarySigma(m) becomes the same, so E.1 and E.2 
and E,3
become the same.

    Yasutoshi



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