[seqfan] Unitary AP Generalized 2

[zbi74583_boat at yahoo.co.jp]

    Hi  Seqfans
    The formula of amicable pair is the following
    Sigma(x) = Sigma(y) = x+y
    More generally  it is possible to describe as follows
    Sigma(x) = Sigma(y) = f(x,y)

    The easiest function of x,y is x+y. If f(x,y) is replaced by more complicated function, it will become an interesting AP like this

http://list.seqfan.eu/pipermail/seqfan/2018-December/018780.html

    More generalization  of Unitary AP :


    [Uni.1]
    {x(n), y(n)} : UitarySigma(x) = UnitarySigma(y) = 7/72*(x^(1/2)+y^(1/2))*(x^(1/2)+5*y^(1/2))
                     If  x = y  then                           = 7/6*x

    {x(n), y(n)} : {2^3*3^3, = }, {2^4*3^3*17, = }, {2^5*3^3*11, = }, {13*2^6*3^4*7^2, 13*2^10*3^2*5^2},....
                     Where {m, = } means {m, m}

    [Uni.2]
    {x(n), y(n)} : UnitarySigma(x) = UnitarySigma(y) = 5/3*(7/2*x^(1/2)-5/2*y^(1/2))^2
                    If  x = y  then                              = 5/3*x

    {x(n), y(n)} : {2^2*3, = }, {2*3^2, = }, {2^6*3*7*13, = }, {13*41*2^10*3^2*5^2, 13*41*2^6*3^4*7^2},.... 



    [Uni.3]
    {x(n), y(n)} : UnitarySigma(x) = UnitarySigma(y) = 7/4*(21*x^(1/2)-20*y^(1/2))^2
                     If  x = y  then                             = 7/4*x


    {x(n), y(n)} :{2^6*3*5*13, = }, {13*17*2^4*5^2*7^2, 13*17*3^2*47^2}, {2^10*3*5^2*7*13*41, = },....



    [Uni.4]
    {x(n), y(n)} : UnitarySigma(x) = UnitarySigma(y) = 3/2*(9*x^(1/2)-8*y^(1/2))^2
                     If  x = y  then                             = 3/2*x


    {x(n), y(n)} : {2, = }, {20, = }, {24, = }, {360, = }, {816, = }, {1056, = }, {12240, = }, {15840, = }, {29120, = }, {181632. = }.
    {5^4*7^2*13*41*79*157*313*2^6*3^4 ,  5^4*7^2*13*41*79*157*313*73^2}



    [Uni.5]


    {x(n), y(n)} : UnitarySigma(x) = UnitarrySigma(y) = 2*(k*x^(1/2)-(k-1)*y^(1/2))*(m*x^(1/2)-(m-1)*y^(1/2))

                                             If  x = y  then     = 2*x




               k = 5, m  = -7
    {x(n), y(n)} : {6, = }, {60, = }, {90, = }, {87360, = }, {5^2*13*2^2*3^2, 5^2*13*7^2}, ....{146361946186458562560000, =}

               k = 2 , m = 41
     {x(n), y(n)} :  {6, = }, {60, = }, {90, = }, {87360, = },
    {5^4*7^2*13*41*79*157*313*2^6*3^4, 5^4*7^2*13*41*79*157*313*73^2}, {146361946186458562560000, =}

               k = 7/2, m = 77/12
    {x(n), y(n)} :  {6, = }, {60, = }, {90, = }, {87360, = }, {13*41*2^10*3^2*5^2, 13*41*2^6*3^4*7^2}. {146361946186458562560000, =}



               k = 9, m = 25
    {x(n), y(n)} :   {6, = }, {60, = }, {90, = }, {87360, = }, {5^4*13*41*79*157*313*2^6*3^4*7^2, 5^4*13*41*79*157*313*7^2*73^2},
    {146361946186458562560000, =}

               k = 91/31, m = 192/31
    {x(n), y(n)} :  {6, = }, {60, = }, {90, = }, {87360, = }, {13*41*2^10*3^2*5^2. 13*41*7^2*73^2}, {146361946186458562560000, =}

               k = 7, m =  6
    {x(n), y(n)} :  {6, = }, {60, = }, {90, = }, {5*17*2^4*3^2, 5*17*13^2}. {87360, = }, {146361946186458562560000, =}

               k = 7, m = -6
    {x(n), y(n)] :  {6, = }, {60, = }, {90, = }, {87360, = }, {3^4*5*17*41*2^2*13^2, 3^4*5*17*41*2^4*7^2}, {146361946186458562560000, =}

               k = 8, m = 8
    {x(n), y(n)}  :  {6, = }, {60, = }, {90, = }, {87360, = }, {3^2*5^3*17*2^2*13^2. 3^2*5^3*17*2^4*7^2}, {146361946186458562560000, =}

               k = 21, m = 36
    {x(n), y(n)}  :  {6, = }, {60, = }, {90, = }, {87360, = }, {13*17*2^4*5^2*7^2, 13*17*3^2*47^2}, {146361946186458562560000, =}

               k = 9, m = -12
    {x(n), y(n)}  :  {6, = }, {60, = }, {90, = }, {87360, = }, {3^2*17*2^2*5^2*13^2, 3^2*17*2^4*5^2*7^2}, 146361946186458562560000, =}

                k = 36/11, 81/11
    {x(n), y(n)}  :  {6, = }, {60, = }, {90, = }, {87360, = }, {17*2^2*5^2*13^2, 17*3^2*47^2}, 146361946186458562560000, =}

    Could anyone confirm them ?



    Yasutpshi