(-1)SSU

[koh]

    Richard Guy
    >
    I am sorry, I should have written the definitions.
    Because few mathematician know (-1) Sigma.

    Richard Mathar
    >  
    Thank you.

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         (-1)SSU GCD-Reduced Amicable Triple : 

         (-1)Sigma(i)*Sigma(i)/UnitaryPhi(i)= k_0*(i+j+k+GCD(i,j,k)) 
         (-1)Sigma(j)*Sigma(j)/UnitaryPhi(j)= k_0*(i+j+k+GCD(i,j,k))
         (-1)Sigma(k)*Sigma(k)/UnitaryPhi(k)= k_0*(i+j+k+GCD(i,j,k))

         k_0=2
         i=2^8*37*73*509*7*59
         i=2^8*37*73*509*19*23
         i=2^8*37*73*509*419

         (-1)SSU Amicable Quadruple : 

         (-1)Sigma(h)*Sigma(h)/UnitaryPhi(h)=h+i+j+k
         (-1)Sigma(i)*Sigma(i)/UnitaryPhi(i)=h+i+j+k
         (-1)Sigma(j)*Sigma(j)/UnitaryPhi(j)=h+i+j+k
         (-1)Sigma(k)*Sigma(i)/UnitaryPhi(i)=h+i+j+k

         h=2*3^2*13*5*11*7*23
         i=2*3^2*13*5*11*191
         j=2*3^2*13*71*7*23
         k=2*3^2*13*71*191

         h=3^2*5^2*13*29*31*7*23*11*19
         i=3^2*5^2*13*29*31*7*23*239
         j=3^2*5^2*13*29*31*191*11*19
         k=3^2*5^2*13*29*31*191*239





    ***********

    I am sure the combination of (-1) Sigma and Sigma and UnitaryPhi has a good structure.   
    Generalizations of Amicable number such that Amicable Triple, Quadruple, Unitary, Reduced, Augment, Rational, Reciprocal are well known.

    http://mathworld.wolfram.com/search/index.cgi?q=kohmoto


    If you replace Sigma(m) in an equation with "(-1)Sigma(m)*Sigma(m)/unitaryPhi(m)",then you get an interesting equation which has as many solutions as the corresponding original one.

    Yasutoshi