(-1)Sigma,Sigma,UnitarySigma,UnitaryPhi

[Richard Mathar]

> From seqfan-owner at ext.jussieu.fr  Wed Sep 27 07:19:46 2006
> Return-Path: <seqfan-owner at ext.jussieu.fr>
> Subject: (-1)Sigma,Sigma,UnitarySigma,UnitaryPhi
> From: "koh" <zbi74583 at boat.zero.ad.jp>
> To: seqfan at ext.jussieu.fr
> ...
>     These two farther generalizations of perfect number are interesting so I think they may fit to UPINT4, if will be published.
> 
>     (-1)Sigma(m)*Sigma(m)/UnitaryPhi(m)=k*m             ....E_1
>     (-1)Sigma(m)*UnitarySigma(m)/UnitaryPhi(m)=k*m      ....E_2
>     
> 
>     S_1 : 2*3, 2^2*5*7, 2^3*3*13, 2^3*3*5*13, 2^4*29*31, 2^5*3*7*61, 2^8*7*19*37*73*509, 2^8*5*7*19*37*509, 2^9*3*11*31*1021, 2^11*3^6*5*7*13*23*137*467*1093*4093
>            K= 2,4,5,6,4,6,5,6,6,7
> 
>     S_2 : 2*3, 2^3*3*13, 2^4*3*17*29, 2^5*3*11*61, 2^8*3*11*43*257*509  
> 
>            K= 2,3,3,3,3
> 
>     I have not done a exhaustive search.
>...
    
The sequence E_1 (also named S_1 above) would start with
6[2, 1; 3, 1]2
140[2, 2; 5, 1; 7, 1]4
312[2, 3; 3, 1; 13, 1]5
1560[2, 3; 3, 1; 5, 1; 13, 1]6
14384[2, 4; 29, 1; 31, 1]4
18018[2, 1; 3, 2; 7, 1; 11, 1; 13, 1]4
40992[2, 5; 3, 1; 7, 1; 61, 1]6
2337400[2, 3; 5, 2; 13, 1; 29, 1; 31, 1]6
7012200[2, 3; 3, 1; 5, 2; 13, 1; 29, 1; 31, 1]8
11027016[2, 3; 3, 4; 7, 1; 11, 1; 13, 1; 17, 1]11

where the first number is m=2*3, 2^2*5*7,..., the stuff in square brackets is
the prime factor decomposition, and the last number is k=2,4,5,6,4,4,6,6...
(also named K above).  This is found with the following PARI program which
exhausts E_1=S_1 up to m=11027016 as shown above:

\\ computes uphi(n)
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) ;
}

\\ computes (-1)sigma(n)
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) ;
}

\\ returns 0 if not in E_1, else k
isA(n)={
	local(s) ;
	s=sigma(n)*A049060(n) ;
	if( s % (n*A047994(n)) == 0,
		return(s/n/A047994(n)) ,
		return(0)
	) ;
}

{
	for(n=1,30000000,
		k=isA(n) ;
		if( k,
			print(n,factor(n),k) 
		) ;
		
	) ;
}

R.J. Mathar, http://www.strw.leidenuniv.nl/~mathar