# [seqfan] Relationship between RMPN and RAN

zbi74583.boat at orange.zero.jp zbi74583.boat at orange.zero.jp
Thu Oct 31 01:40:39 CET 2013

```    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

```