[seqfan] Uphi(m)=UPhi(n)=3/4*(m*n)^(1/2)

koh zbi74583.boat at orange.zero.jp
Tue Nov 4 07:20:21 CET 2008


    Hi, Seqfans
    Definition of Amicable Number is the following.

    Sigma(x) = Sigma(y) = x+y

    If "Sigma" is replaced with the other divisor functions or "x+y" is replaced with high degree or rational or irrational formulas then it becomes generalized Amicable Number.

    I calculated the examples of the following equtation.    

    UnitaryPhi(x) = UnitaryPhi(y) = 3/4*(x*y)^(1/2) , y<=x

    x=y=2^2
    x=y=2^3*7
    x=y=2^4*5
    x=y=2^5*5*31
    x=y=2^6*3^2*7
    x=y=2^6*3^3*7*13
    x=y=2^7*3^2*7*127
    x=y=2^7*3^3*7*13*127
    x=y=2^8*5*17          
    x=y=2^9*3^2*7*73
    x=y=2^9*3^3*7*13*73
    x=y=2^10*3^3*5^2*11*13*31
               
    x=5*3^2*11^2,      y=5*31^2 
    x=2^11*5*3^2*89^2, y=2^11*5*11^2*23^2

    Could anyone do exhausitive search?
    I would like to know the other examples of the case not{x=y}.




    %I A000001
    %S A000001 4, 56, 80, 4032, 4960, 5445, 21760, 157248, 1024128, 39940992, 2354688, 91832832, 729999360
    %N A000001 Numbers m,n such that UnitaryPhi(m)=UnitaryPhi(m)=3/4*(m*n)^(1/2), n<=m
    %C A000001 a(6) and a(13) are the cases not{n=m}  
    %e A000001 Factorization of cases not{n=m}
               m,n=5*3^2*11^2,5*31^2
               m.n=2^11*5*3^2*89^2,2^11*5*11^2*23^2
    %Y A000001 A000002
    %K A000001 none
    %O A000001 0,1
    %A A000001 Yasutsohi Kohmoto zbi74583.boat at orange.zero.jp

    %I A000002
    %S A000002 4, 56, 80, 4032, 4960, 4805, 21760, 157248, 1024128, 39940992, 2354688, 91832832, 655452160

    %N A000002 Numbers m,n such that UnitaryPhi(m)=UnitaryPhi(m)=3/4*(m*n)^(1/2), n<=m
    %C A000002 a(6) and a(13) are the cases not{n=m}  
    %Y A000002 A000001
    %K A000002 none
    %O A000002 0,1
    %A A000002 Yasutsohi Kohmoto zbi74583.boat at orange.zero.jp



    Yasutoshi
    




More information about the SeqFan mailing list