(-1)SSU

[Richard Guy]

Please define (-1)SSU
               (-1)Sigma(n)
               UnitaryPhi(n)
R.

On Sat, 28 Oct 2006, koh wrote:

>    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("") ;
> ) ;
> ) ;
>
> ) ;
> }
>