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<DIV><FONT face="MS UI Gothic" size=2><FONT size=3> Hello,
Seqfans.</FONT></FONT></DIV>
<DIV><FONT face="MS UI Gothic"></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"> I considered Gaussian Amicable
Pair.</FONT></DIV>
<DIV><FONT face="MS UI Gothic"> It is defined as
follows.</FONT></DIV>
<DIV><FONT face="MS UI Gothic"></FONT> </DIV>
<DIV><FONT face="MS UI Gothic">
[Gaussian AP]</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2><FONT size=3></FONT></FONT> </DIV>
<DIV><FONT face="MS UI Gothic" size=2><FONT
size=3>
GSigma(x)=GSigma(y)=Fq(m*(u*Fq(x)+Fq(y)) , for some Gaussian integer
m - GA -<BR><BR> where, u means unit.
<BR> If n=Product p_i^r_i then GSigma(n)=Fq(Product(Sum Fq(p_i^s_i) ,
0<=s_i<=r_i)) <BR><BR>
Fq(n)=i^k*n , 0<=k<=3
<BR> If Fq(n) is of the form
r*e^(i*t) then 0<=t<Pi/2 </FONT></FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2><FONT size=3></FONT></FONT> </DIV>
<DIV><FONT face="MS UI Gothic" size=2><FONT
size=3> e.g.
Fq(-1+4i)=i^3*(-1+4i)=4+i</FONT></FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2><FONT size=3></FONT></FONT> </DIV>
<DIV><FONT face="MS UI Gothic" size=2><FONT
size=3>
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.
</FONT></FONT></DIV>
<DIV><FONT face="MS UI Gothic"></FONT> </DIV>
<DIV><FONT
face="MS UI Gothic">
Total[n]=1/4*Sum{ Fq(d) , d|n</FONT> <FONT
face="MS UI Gothic">}</FONT></DIV>
<DIV><FONT face="MS UI Gothic"></FONT> </DIV>
<DIV><FONT face="MS UI Gothic" size=2><FONT
size=3>
But I like this function. So, I used Fq(n) for defining Gaussian
AP.</FONT></FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2><FONT
size=3> <BR></FONT></FONT><FONT
face="MS UI Gothic" size=2><FONT size=3></FONT></FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2><FONT size=3> Examples of GA :
u=-i <BR></FONT></FONT><FONT face="MS UI Gothic" size=2><FONT
size=3></FONT></FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2><FONT size=3></FONT></FONT> </DIV>
<DIV><FONT face="MS UI Gothic" size=2><FONT
size=3> x=(1+i)^11*(1+2i)*(2+i)*7*359
<BR> y=(1+i)^11*(1+2i)*(2+i)*7*(5+6i)*(29+30i)
<BR> m=(1+i)^2*(1+2i)*(2+i) </FONT></FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2><FONT size=3></FONT></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"></FONT> </DIV>
<DIV><FONT face="MS UI Gothic" size=2><FONT
size=3> x=(1+i)^11*(1+2i)*(2+i)*3^5*7^2*19*431 <BR>
y=(1+i)^11*(1+2i)*(2+i)*3^5*7^2*19*(5+6i)*(35+36i) </FONT></FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2><FONT
size=3>
m=(1+i)^6</FONT> <FONT size=3> </FONT></FONT></DIV>
<DIV><FONT face="MS UI Gothic"> </FONT></DIV>
<DIV><FONT face="MS UI Gothic"> I made an easy algorithm for
the equation GA which make a calculation on Z[i] to be almost a calculation on
Z. </FONT></DIV>
<DIV><FONT
face="MS UI Gothic">
</FONT></DIV>
<DIV><FONT face="MS UI Gothic"> For submitting this sequence,
I should do a computer search, but I don't know how to write the GSigma(n)
in MATHEMATICA.</FONT></DIV>
<DIV><FONT face="MS UI Gothic"> I wish someone tell me
it. </FONT></DIV>
<DIV><FONT face="MS UI Gothic"> For a computer, the following
equation is much easier. </FONT></DIV>
<DIV><FONT face="MS UI Gothic">
Sigma(x)=Sigma(y)=x+y , x,y are Gaussian integer.</FONT></DIV>
<DIV><FONT face="MS UI Gothic">
If n=Product p_i^r_i then Sigma(n)=Product
(p_i^(r_i+1)-1)/(p_i-1) </FONT></DIV>
<DIV><FONT face="MS UI Gothic"> </FONT></DIV>
<DIV><FONT face="MS UI Gothic"> Yasutoshi</FONT></DIV>
<DIV><FONT face="MS UI Gothic"> </FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2>
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