[seqfan] Re: Relationship between RMPN and RAN

[M. F. Hasler]

Cf. A160320 (sigma(n)/n=k+1/3), A160321 (sigma(n)/n=k+2/3), A159907
(sigma(n)/n=k+1/2).

See also http://www.numericana.com/answer/numbers.htm#multiperfect
(for half-integral values)

Maximilian

On Wed, Oct 30, 2013 at 8:40 PM,  <zbi74583.boat at orange.zero.jp> wrote:
>     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
>
>
>
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>
>
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