Gaussian Amicable Pair

[kohmoto]

    Hello, Seqfans.


    I considered Gaussian Amicable Pair.
    It is defined as follows.

         [Gaussian AP]

         GSigma(x)=GSigma(y)=Fq(m*(u*Fq(x)+Fq(y)) , for some Gaussian 
integer m      - GA -

　　　　where, u means unit.
　　　　If n=Product p_i^r_i then GSigma(n)=Fq(Product(Sum Fq(p_i^s_i) , 
0<=s_i<=r_i))

　　　　         Fq(n)=i^k*n , 0<=k<=3
　　　　         If Fq(n) is of the form r*e^(i*t) then 0<=t<Pi/2

 　              e.g.   Fq(-1+4i)=i^3*(-1+4i)=4+i

                 Comment : It is not necessary to use Fq(n) for defining 
equations like Gaussian AP, though the sum of divisors of a Gaussian integer 
is defined with Fq(n) as follows. "Fq" is for first quadrant.

                         Total[n]=1/4*Sum{ Fq(d) , d|n }

                 But I like this function. So, I used Fq(n) for defining 
Gaussian AP.


　　　　Examples of GA :  u=-i


　　　　x=(1+i)^11*(1+2i)*(2+i)*7*359
　　　　y=(1+i)^11*(1+2i)*(2+i)*7*(5+6i)*(29+30i)
　　　　m=(1+i)^2*(1+2i)*(2+i)


         x=(1+i)^11*(1+2i)*(2+i)*3^5*7^2*19*431
         y=(1+i)^11*(1+2i)*(2+i)*3^5*7^2*19*(5+6i)*(35+36i)
         m=(1+i)^6

    I made an easy algorithm for the equation GA which make a calculation on 
Z[i] to be almost a calculation on Z.

    For submitting this sequence, I should do a computer search, but I don't 
know how to write the GSigma(n) in MATHEMATICA.
    I wish someone tell me it.
    For a computer, the following equation is much easier.
         Sigma(x)=Sigma(y)=x+y , x,y are Gaussian integer.
         If n=Product p_i^r_i then Sigma(n)=Product (p_i^(r_i+1)-1)/(p_i-1)

    Yasutoshi

 
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://list.seqfan.eu/pipermail/seqfan/attachments/20050308/25f308b3/attachment.htm>