[seqfan] Semi group like system on A_{m,n}

[zbi74583.boat at orange.zero.jp]

    Hi, Seqfans


    I explain the reason why I added the condition If i=1 Then k=1 on the
product (i,j)*(k,l) of A050521

    It is an abstruction of the Algebla of A_{m,n} which is generalized Amicable
n-tuple.

    The menbers of the set of all A_{m,n} make a semi group like system.

    [ Definition ]
    A_{m,n} is n-tuple which satisfy the following

    Sigma(x_i)=Sum_{1<=j<=n} m/n*x_j   1<=i<=n ....-E0-

    If m=n then it become  Amicable n-tuple.
    So, it is a generalization of A_n.

    If m=1 then

    Sigma(x_i)=Sum_{1<=j<=n} 1/n*x_j
              <=x_0
          Where x_0 is Max(x_i)
    Only one solution x_0=1
    So,   x_i=1   1<=i<=n
    Hense n=1
    It means 1<m,1<=n except A_{1,1}


    [ Product of A_{m,n} ]

    A_{m,n}*A_{r,s}=A_{m*r,n*s}

    It means the following

    Sigma(x_i)*Sigma(y_k)=Sigma(x_i*y_k)   1<=i<=n  1<=k<=s ....-E1-

   (Sum_{1<=j<=n} m/n*x_j)*(Sum_{1<=l<=s} r/s*y_l)
    =Sum_{1<=j<=n 1<=l<=s} m/n*r/s*x_j*y_l ....-E2-
    Where GCD(x_i.y_k)=1   1<=i<=n 1<=k<=s ....-C0-

    From E0,E1 and E2, we obtein the definition of A_{m*r,n*s}
    The unit is A_{1,1} so it is semi group with the C0

    [ Example ]
    A_{4,4} is factorized as follows
    A_{4,4}=A_{2,1}*A_{2,4}=A_{2,2}*A_{2,2}= A_{2,4}*A_{2,1}
    So A051707(4)=3
    These factorizations give the method how to make A_{4,4}

    http://mathworld.wolfram.com/AmicableQuadruple.html
    The example on this page is A_{2,1}*A_{2,4}
    The method  on this page is A_{2,2}*A_{2,2}


    [ Comment ]

    The m of A_{m,n} represents its size.
    Because  A_{m,1} is m Multiply Perfect Number.
    The size of A_{m,n} is almost the same as A{m,1}
    For example A_{11,1} has 1900  digits which is known only one.
    So  probably A_{11,n} has at least 1900  digits.

    I wonder why Neil wrote the keyword "nice".
    Does something interesting exist on the algebla?
    Does any essencial difference between A051707 and A108462?


    Yasutoshi