[seqfan] Relationship between RMPN and RAN

[zbi74583.boat at orange.zero.jp]

    Hi, Seqfans

    [ Relationship between Rational Multiply Perfect Number and Rational
Amicable Number ]

    [ Definition ]

    RMPN :
    Sigma(m)=k*m , where k is Rational Number

    RAN :
    Sigma(m)=Sigma(n)=F(m,n) , where F(m,n) is Rational formula of m,n

    http://mathworld.wolfram.com/RationalAmicablePair.html

    [ How to solve ]

    Example ;

    Sigma(x)=Sigma(y)=(x+y)^3/(x^2+y^2)    ....E0

    Let
    x=c*n
    y=c*m
    GCD(c,n)=1 , GCD(c.m)=1

    From E0
    Sigma(c*n)=Sigma(c*m)
    Hence
    Sigma(n)=Sigma(m)    - E1 -

    From E0
    Sigma(c*n)=c*(n+m)^3/(n^2+m^2)
    Hence
    Sigma(c)=k*c , k=(n+m)^3/((n^2+m^2)*Sigma(n))    - E2 -
    c is RMPN

    If we try with
    n=33 , m=47

    It satisfies E1 , because
    Sigma(3*11)=48
    Sigma(47)=48

    k=80^3/(2*17*97*3*2^4)
     =2^7*6^3/2*17*97

    We compute c
    c=2^2*3^3*7^3*17*97

    Unfortunatly
    GCD(c,n)=3 is not 1
    So, (c*n, c*m) is not RAN


    If we try with (n,m) which is not smooth and n+m is smooth then (c*n, c*m)
becomes RAN
    Where n is smooth  means all prime factors of n are small
    Where m is small  means m<100

    To compute RMPN is neccesary for  solving RAN.
    So, RMPN is important and interesting problem.

    I am going to compute all RMPN such that k=n/m<7 and m<100.
    I computed these RMPN in my hand, exactly in my mind.

    Could anyone confirm them and compute their decimal expression?
    If you found the other solution, then tell me it.

    [ RMPN k=m/3 4<=m<=19 ]

    4/3
    3

    7/3
    2^2*3
    2*3^2*13

    8/3
    3*2^(m-1)*M_m
              Where M_m means Mersenne prime 2^m-1
    2*3^2*7*13
    2*3^3*5

    10/3
    2^3*3^2*7*13
    2^3*3^3*5
    2^7*3^6*17*23*137*547*1093
    2^25*3^6*19*23*137*547*683*1093*2731*8191
    2^13*3^3*11*43*127

    11/3
    2^3*3^4*5*11
    2^5*3^4*7*11
    2^9*3^2*7*13*31
    2^9*3^3*5*31
    2^13*3^2*7*13*43*127
    2^13*3^3*5*43*127

　　13/3
    2^8*3^2*5*7*19*37*73
    2^14*3^2*5*7*19*31*151
    2^11*3^3*5^2*7*31
    2^11*3^6*5*7*23*137*547*1093
    2^7*3^5*5*7*17
    2^10*3^5*5*7*23*89
    2^25*3^5*5*7*19*2731*8191

    14/3
    2^5*3^4*5*7*11^2*19
    2^7*3^4*5^2*11^2*17*19*31
    2^10*3^4*5^2*11^2*23*19*31*89
    2^11*3^3*5^2*7*13*31
    2^8*3^2*5*7*13*19*37*73
    2^14*3^2*5*7*13*19*31*151
    2^7*3^7*5^2*11*17*31*41*43
    2^17*3^7*5*7*19^2*37*41*73*127
    2^11*3^6*5*7*13*23*137*547*1093
    2^5*3^5*5*7^2*13*19
    2^7*3^5*5*7*13*17
    2^10*3^5*5*7*13*23*89
    2^11*3^5*5*7*13^2*31*61
    2^25*3^5*5*7*13*19*2731*8191

    16/3
    2^15*3^7*5^2*7*11*17*31*41*43*257
    2^7*3^4*5^2*7*11^2*17*19*31
    2^7*3^5*5^2*7^2*13*17*19*31
    2^10*3^4*5^2*7*11^2*19*23*31*89
    2^10*3^5*5^2*7^2*13*19*23*31*89
    2^13*3^5*5*7^2*11*13*19*43*127

    17/3
    2^15*3^5*5^2*7^2*11*13*19*31*43*257

    19/3
    2^24*3^8*5^2*7*11*13^2*17*19*31^3*37*43*53*61*379*601*757*1801
    Only this one I used my note book computer.

    Neil  you may find many sequences in this result.



    Yasutoshi