(-1)SSU

[koh]

    More examples of (-1)SSU Amicable Number :

         k=3
         2^9*3*31*1021*7*23
         2^9*3*31*1021*191    

         k=2
         3^2*5^2*13*29*17*19
         3^2*5^2*13*29*359    



    (-1)SUU Amicable Number : 

         (-1)Sigma(m)*USigma(m)/UnitaryPhi(m)=k(m+n) 
         (-1)Sigma(n)*USigma(n)/UnitaryPhi(n)=k(m+n) , for some integer k.

         k=1
         3^2*5*7*23
         3^2*5*191             

         k=1
         3*5^2*13*29*11*!9
         3*5^2*13*29*239      


          
                 
    (-1)SUU  GCD-Augment Amicable Triple : 

         (-1)Sigma(i)*USigma(i)/UnitaryPhi(i)=i+j+k-GCD(i,j,k) 
         (-1)Sigma(j)*USigma(j)/UnitaryPhi(j)=i+j+k-GCD(i,j,k) 
         (-1)Sigma(k)*USigma(k)/UnitaryPhi(k)=i+j+k-GCD(i,j,k) 
             
         2^2*3*11*29
         2^2*3*17*19
         2^2*3*359             

    Yasutoshi
    
 



> From seqfan-owner at ext.jussieu.fr  Sun Oct 22 08:51:00 2006
> Return-Path: <seqfan-owner at ext.jussieu.fr>
> Date: Sun, 22 Oct 2006 15:33:31 +0900
> Subject: (-1)SSU
> From: "koh" <zbi74583 at boat.zero.ad.jp>
> To: seqfan at ext.jussieu.fr
> 
>     Hi, Seqfans
> 
>     I calculated (-1)SSU Amicable Number.
> 
>     (-1)Sigma(m)*Sigma(m)/UnitaryPhi(m)=k(m+n) 
>     (-1)Sigma(n)*Sigma(n)/UnitaryPhi(n)=k(m+n) , for some integer k.
> 
>     
> 
>          k=3
>          m=2^5*3*61*5*11
>          n=2^5*3*61*71
> 
>          m=2^8*7*37*73*509*3*5
>          n=2^8*7*37*73*509*23
>          
>          m=2^8*7*19*37*73*509*3*11
>          n=2^8*7*19*37*73*509*47
> 
>          m=2^8*5*7*37*73*509*11*59
>          n=2^8*5*7*37*73*509*719
> 
> 
> 
>          k=2
>          m=2*3^2*5*13*23*29
>          n=2*3^2*5*13*719
> 
>          m=2^2*7*3*11
>          n=2^2*7*47
> 
> 
> 
>          k=5
>          m=2^5*3^2*7*13*61*23*29
>          n=2^5*3^2*7*13*61*719
>          
>          m=2^9*3^2*11*13*31*1021*23*29
>          n=2^9*3^2*11*13*31*1021*719
> 
> 
> 
>     I wish some one verify them.
> 
>     Yasutoshi

All of them up to n<=m<=1560 are:

k=1
m=6 [2, 1; 3, 1]
n=6 [2, 1; 3, 1]

k=2
m=140 [2, 2; 5, 1; 7, 1]
n=140 [2, 2; 5, 1; 7, 1]

k=2 (already listed above....)
m=1316 [2, 2; 7, 1; 47, 1]
n=924 [2, 2; 3, 1; 7, 1; 11, 1]

k=3
m=1560 [2, 3; 3, 1; 5, 1; 13, 1]
n=1560 [2, 3; 3, 1; 5, 1; 13, 1]

and generally each A122483 which is associated with
an even k' in A123124 gives a "degenerate" solution of these (-1)SSU pairs
with k=k'/2 and m=n. So what I am listing above as m=n=6, m=n=140 and m=n=1560
is also found in A122483 .

RJM

In PARI the generating program is as follows (and this also verfies that
the 6 cases given above all obey the equation):
\\ unitary phi
A047994(n)={
local(i,resul,rmax) ;
if(n==1,
return(1)
) ;
i=factor(n) ;
rmax=matsize(i)[1] ;
resul=1 ;
for(r=1,rmax,
resul *= i[r,1]^i[r,2]-1 ;
) ;
return(resul) ;
}

\\ (-1)sigma
A049060(n)={
local(i,resul,rmax,p) ;
if(n==1,
return(1)
) ;
i=factor(n) ;
rmax=matsize(i)[1] ;
resul=1 ;
for(r=1,rmax,
p=0 ;
for(j=1,i[r,2],
p += i[r,1]^j ;
) ;
resul *= p-1 ;
) ;
return(resul) ;
}

\\ return k if a valid pair, or 0 if not.
isssu(m,n)={
local(pm,um,k,pn,un) ;
pm=A049060(m)*sigma(m) ;
um=(m+n)*A047994(m) ;
if( pm % um == 0,
k=pm/um ;
pn=A049060(n)*sigma(n) ;
un=(m+n)*A047994(n) ;
if ( pn == k*un,
return(k),
return(0)
) ,
return(0)
) ;
}

{
\\ test all cases already known..
        m=2^5*3*61*5*11 ;
        n=2^5*3*61*71 ;
        print(isssu(m,n)) ;

        m=2^8*7*37*73*509*3*5 ;
        n=2^8*7*37*73*509*23 ;
        print(isssu(m,n)) ;

        m=2^8*7*19*37*73*509*3*11 ;
        n=2^8*7*19*37*73*509*47 ;
        print(isssu(m,n)) ;

        m=2^8*5*7*37*73*509*11*59 ;
        n=2^8*5*7*37*73*509*719 ;
        print(isssu(m,n)) ;

        m=2*3^2*5*13*23*29 ;
        n=2*3^2*5*13*719 ;
        print(isssu(m,n)) ;

        m=2^5*3^2*7*13*61*23*29 ;
        n=2^5*3^2*7*13*61*719 ;
        print(isssu(m,n)) ;

        m=2^9*3^2*11*13*31*1021*23*29 ;
        n=2^9*3^2*11*13*31*1021*719 ;
        print(isssu(m,n)) ;

\\ systematic search
for(m=1,3000000000,
for(n=1,m,
k=isssu(m,n) ;
if(k,
print("k=",k) ;
print("m=",m," ",factor(m)) ;
print("n=",n," ",factor(n)) ;
print("") ;
) ;
) ;

) ;
}