(-1)Sigma,Sigma,UnitarySigma,UnitaryPhi

[Richard Mathar]

> From seqfan-owner at ext.jussieu.fr  Wed Sep 27 07:19:46 2006
> Date: Wed, 27 Sep 2006 14:02:33 +0900
> Subject: (-1)Sigma,Sigma,UnitarySigma,UnitaryPhi
> From: "koh" <zbi74583 at boat.zero.ad.jp>
> To: seqfan at ext.jussieu.fr
> 
>     Dear Guy and Seqfans.
> 
>     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.
>     I wish someone will do it.
> 
>     Yasutoshi

S_2 (also named E_2 above) starts as

n=6 (k=2)
n=312 (k=3)
n=495 (k=2)
n=990 (k=3)
n=20520 (k=4)
n=23664 (k=3)
n=64416 (k=3)
n=13063050 (k=4)

and there are no more terms smaller than n=74000000.

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