(-1)SSU

[Richard Mathar]

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