S_{2,1}

[koh]

    [Definition of  {k,l}-Aliquot sequence]

    Let S_{k,l}(n)=1/l*Sigma(n)-k*n .

    b(n)=S_{k,l}(b(n-1)) 

    If b(n)=b(n+m) then it is called "{k,l}-Aliquot cycle" or "{k,l}-Sociable number of order m".
    I submit an example of {k,l}-Aliquot cycle.
    
    This is  S_{2,1} .


    Yasutoshi

    ----------
 
    %I A000001
    %S A000001 51, 72, 120, 132, 672, 2602, 4750, 10054, 14884, 45840, 51168
    %N A000001 Let S(n)=Sigma(n)-2*n .
                     Numbers such that S(S(S(S(n))))=n, {2,1}-Sociable number of  order 1 or 2 or 4. .
    %C A000001 Each cycle has some negative integers as members. 
                     If n is a negative integer then Sigma(n)=-Sigma(-n) .
                     Orders of each cycle are 4,4,1,4,1,4,4,4,4,2,2.
    %e A000001 {51,-30,132,72} is a {2,1}-Aliquot cycle.
    %H A000001 <a href="http://mathworld.wolfram.com/SociableNumbers.html">WathWorld</a>  
    %Y A000001 A113791
    %K A000001 none
    %O A000001 1,1
    %A A000001 Yasutsohi Kohmoto   zbi74583 at boat.zero.ad,jp 

    ----------

    The following early posting doesn't appear on OEIS.
    They are S_{0,4} and S_{0,3}.

    quote :

    %I A000001
    %S A000001 30240, 32760, 859320, 898560
    %N A000001 Let S(n)=1/4*Sigma(n) .
               Numbers such that S(S(n))=n, 1/4-Sociable number of  order 1 or 2.
    %C A000001 a(1) and a(2) are 4 multiple perfect numbers. a(4)=S(a(3)). 
               I calculated {2096640,2234232}. It is unknown if any pair exists between {a(3),a(4)} and this pair.
    %H A000001 <a href="http://mathworld.wolfram.com/SociableNumbers.html">WathWorld</a>  
    %Y A000001 A000002
    %K A000001 none
    %O A000001 1,1
    %A A000001 Yasutsohi Kohmoto   zbi74583 at boat.zero.ad,jp 


    %I A000002
    %S A000002 120, 672, 7680, 8184, 523776
    %N A000002 Let S(n)=1/3*Sigma(n) .
               Numbers such that S(S(n))=n, 1/3-Sociable number of order 1 or 2.
    %C A000002 a(1) and a(2) and a(5) are 3 multiple perfect numbers. a(4)=S(a(3)). 
    %H A000002 <a href="http://mathworld.wolfram.com/SociableNumbers.html">WathWorld</a>  
    %Y A000002 A000001 
    %K A000002 none
    %O A000002 1,1
    %A A000002 Yasutoshi Kohmoto   zbi74583 at boat.zero.ad,jp