Gaussian Perfect numbers on Z[i]

[T. D. Noe]

Except for the first two, these seem very artificial.  For Sigma(n), the
Gaussian primes must be in the first quadrant.

>    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.

-- 

Tony Noe                  | voice:     503-690-2099
Software Spectra, Inc.    | fax:       503-690-8159
14025 N.W. Harvest Lane   | e-mail:    noe at sspectra.com
Portland, OR  97229, USA  | Web site:  http://www.sspectra.com