Sigma_2(n)

[y.kohmoto]

    Hello, seqfans.
    I searched small solutions of the following equations. x<100000.
    Sigma_2(x)=Sigma_2(y)=x^2+y^2 
    Sigma_2(x)=k*x^2
    Where Sigma_2(n) means the sum of square of divisors of n.
    But if my software is correct, no solution exists.
    Is it possible to prove it? Or, does a solution exist?
    

    I knew the following sequence doesn't exist on OEIS.
    a(n) : 1,4,9,16,25,36,49,50,64,81,100,117,
         Sigma_2(n)/Sigma(n)=integer 
    example : 
         Sigma_2(4)/Sigma(4)=(1+4+16)/(1+2+4)=3 , so 4 appears on a(n).
    It is easy but interesting.
    I will try to calculate Sigma_i(n)/Sigma_j(n) on several pairs {i,j}.

    Yasutoshi