(-1)Sigma,Sigma,UnitarySigma,UnitaryPhi

[kohmoto]

    Hi, Richard Mathar

    Thank you for a search with a computer.
    I will submit it to OEIS.

    I will describe that Richard Mathar calculated some terms.


    Neil
    >Me:   would one of you please submit it to the OEIS (unless you did 
already)
    Yes.

    Yasutoshi


----- Original Message ----- 
From: "Richard Mathar" <mathar at strw.leidenuniv.nl>
To: <seqfan at ext.jussieu.fr>
Sent: Thursday, September 28, 2006 2:44 AM
Subject: Re: (-1)Sigma,Sigma,UnitarySigma,UnitaryPhi


>
>> 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
>