Gaussian Perfect numbers on Z[i]

[y.kohmoto]

    [Collection of divisor functions for Gaussian integer ]
    Total(n) :
         The sum of divisors of n.
         Use the first quadrant rule.
         No formula exists.
         It doesn't have the property of multiplicative.

    Sigma(n) :
         Product  (p_i^( _i+1)-1)/(p_i-1) ....Spira
         Where n=Product p_i^ _i
         Don't use the first quadrant rule, when you calculate a difference.

    GSigma(n) :
         Product (Sum(p_i^s_i , 0<=s_i<=r_i)) ....Kohmoto
         Where n=Product p_i^ r_i
         Use the first quadrant rule, when you calculate a sum.

    NSigma(n) :
         the sum of the norms of the divisors. ....Franklin T. Adams-Watters
         What is the formula?

    I would like to define more functions.

    UnitarySigma(n) :
         Product (p_i^r_i + 1)
         Don't use the first quadrant rule, when you calculate a sum

    +1UnitarySigma(n) :
         Product (p_i^r_i + 1)
         Use the first quadrant rule, when you calculate a sum.

    +iUnitarySigma(n) :
         Product (p_i^r_i + i)
         Use the first quadrant rule, when you calculate a sum.

    -1UnitarySigma(n) or UnitaryPhi(n) :
         Product (p_i^r_i - 1)
         Use the first quadrant rule, when you calculate a difference.

    -iUnitarySigma(n) or +iUnitaryPhi(n) :
         Product (p_i^r_i - i)
         Use the first quadrant rule, when you calculate a difference.
     NSigma_2(n) :
          Product {Sum { Norm(p_i^s_i)^2 , 0<=s_i<=r_i}}

          If Norm(m)=Norm(n) then they are counted as one divisor.
          It becomes a integer function.

          S1 : 1, 3, 6, 18, 18, 42, 38, 93, 84, 126, 102
          Name : NSigma_2(n+k*i) . Second line of an array. k=1
          S0 : 0, 1, 7, 10, 31, 31, 70, 50, 127, 91, 217
          Name : NSigma_2(n+k*i) . The first line of an array. k=0


    [Gaussian Perfect numbers]
    If m is not 2,  then we should add "m-multiple"

    Total Gaussian Perfect number :
    Total(n)=m*n , for some Gaussian integer m
    Example
    (1+i)^5*7                m=1+3i


    Perfect Gaussian integer :          ...McDaniel, Wayne L..
    Sigma(n)=m*n , for some Gaussian integer m
    Example
    (1+i)^5*(2+3i)*3           m=2+i


    Gaussian Perfect number :
    GSigma(n)=m*n , for some Gaussian integer m
    Example
    (1+i)^5*7*(2+i)*(1+2i)   m=(1+i)^5=4+4i


    Unitary Perfect Gaussian integer :
    UnitarySigma(n)=m*n , for some Gaussian integer m
    Example
    (1+i)^5*(2+i)^2          m=1


    Unitary Gaussian Perfect number :
    +1UnitarySigma(n)=m*n , for some Gaussian integer m
    Example
    (1+i)^5*(5+4i)*(3+2i)*(2+i)*(1+2i)         m=(1+i)^3=2+2i


    iUnitary Gaussian Perfect number :
    +iUnitarySigma(n)=m*n , for some Gaussian integer m
    Example
    (1+i)^5*(4+5i)*(2+3i)*(1+2i)*(2+i)         m=(1+i)^3=2+2i


    -1Unitary Gaussian Perfect number :
    -1UnitarySigma(n)=1/m*n , for some Gaussian integer m
    Example
    (1+i)^5*(2+i)^2*(1+2i)   m=1+i


    -iUnitary Gaussian Perfect number :
    -iUnitarySigma(n)=m*n , for some Gaussian integer m
    Example
    (1+i)^5*(1+2i)^2* (2+i)  m=1+i


    I calculated the solutions of the case that (1+i)^5*k , GCD(1+i,k)=1



    I will soon post these  sequences to OEIS.

    Yasutoshi