Sigma,USigma,UPhi 2

[koh]

    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*,357,418,570,714,910,1045,1254,2090,2730,....

    S_2 : 
        1,2,3,6,14,15,30,35,42,70,78,105,190,210,357,418,570,714,910,1045,1254,1976*,2090,2730,....

    S_3 : 
        1,2,3,6,14,15,30,35,42,70,78,105,190,210,312*,357,418,570,714,910,1045,1254,1428*,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