Gaussian Perfect number

[y.kohmoto]

    Neil,
    I post sequences of Gaussian PN to OEIS.


    %I A000001
    %S A000001  1, 5, 10, 10, 30, 30, 140, 200
    %N A000001 Gaussian Perfect number.
                     Definition : GSigma(n)=m*n , for some Gaussian integer
m.
                     Where If n=Product p_i^r_i then GSigma(n)=Product (Sum
p_i^s_i, 0<=s_i<=r_i)
                      Sequence gives real part of Gaussian Perfect number.
    %C A000001      The following condition is  necessary for defining "Sum"
of Gaussian integer.

                      If a Gaussian integer is of the form : r*e^(i*t), then
0<=t<Pi/2.

                      GSigma(n) is a formal sum of divisors of n.
                      Indeed, sometimes it gives a kind of difference of
divisors of n.
     %e A000001  GSigma((1+i)^7)=1+1+i+2+2+2*i+4+4+4*i+8+8+8*i=30+15*i

                       GSigma((1+i)*(2+i))=1+1+i+2+i+1+3i = 5+5i
=(1+i)*(2+i)*(1+2i) , so (1+i)*(2+i)=1+3i is a Gaussian Perfect number.

    %Y A000001 A000002
    %K A000001 nonn
    %O A000001 1,2
    %A A000001 Yasutoshi Kohmoto (zbi74583 at boat.zero.ad.jp)

    %I A000002
    %S A000002  3, 5, 10, 28, 30, 84, 140, 600
    %N A000002 Gaussian Perfect number.
                     Definition : GSigma(n)=m*n , for some Gaussian integer
m.
                     Where If n=Product p_i^r_i then GSigma(n)=Product (Sum
p_i^s_i, 0<=s_i<=r_i)
                      Sequence gives imaginary  part of Gaussian Perfect
number.
    %C A000002      The following condition is  necessary for defining "Sum"
of Gaussian integer.

                       If a Gaussian integer is of the form : r*e^(i*t),
then 0<=t<Pi/2.

                       GSigma(n) is a formal sum of divisors of n.
                       Indeed, sometimes it gives a kind of difference of
divisors of n.
     %e A000002  GSigma((1+i)^7)=1+1+i+2+2+2*i+4+4+4*i+8+8+8*i=30+15*i

                       GSigma((1+i)*(2+i))=1+1+i+2+i+1+3i = 5+5i
=(1+i)*(2+i)*(1+2i) , so (1+i)*(2+i)=1+3i is a Gaussian Perfect number.

    %Y A000002 A000001
    %K A000002 nonn
    %O A000002 1,1
    %A A000002 Yasutoshi Kohmoto (zbi74583 at boat.zero.ad.jp)


    %I A000003
    %S A000003 1, 2, 3, 3, 4, 4, 4, 1
    %N A000003 Real part of m number of Gaussian Perfect number
    %Y A000003 A000001
    %K A000003 nonn
    %O A000003 1,2
    %A A000003 Yasutoshi Kohmoto (zbi74583 at boat.zero.ad.jp)


    %I A000004
    %S A000004  2, 2, 3, 0, 4, 0, 4, 5,
    %N A000004 Imaginary part of m number of Gaussian Perfect number
    %Y A000004 A000001
    %K A000004 nonn
    %O A000004 1,1
    %A A000004 Yasutoshi Kohmoto (zbi74583 at boat.zero.ad.jp)


    I am not sure if the sequence is complete up to r=40^(1/2)*100.
    I wish Farideh will do  exhaustive search.

    GSigma(n) is multiplicative, so this "Gaussian Perfect number" is
interesting..
    But I want to know a formula for "true" sum of divisors of n.

    Example :
　　n=(1+i)^2 * (1+2i)

　　GSigma(n)=(1+1+i+2) * (1+1+2i)= (4+i) * 2* (1+i)=  6+10i
    If all divisors are described, then it becomes as follows.
　　GSigma(n)=1+1+i+2 + 1+2i-1+3i+2+4i=6+10i
    The divisor " 3+i " is multiplied by the unit "i" , so it is a kind of
formal sum of divisors of n.

    The "Sum" of divisors of n is the following.
    TGSigma(n)=1+1+i+2 + 1+2i+3+i+2+4i= 10+8i
               Where -1+3i == 3+i   mod Pi/2

    Yasutoshi