COMMENT on A113285.

[koh]

    Hi, Bob
    Thanks for calculating more terms.

    I think your calculation is a nice work.
    Because I have conjectured that proper {2,1} Aliquot cycles of order 4 must exist, but I supposed that they have much more digits than terms which I calculated.

    Where "proper Aliquot cycle" means Aliquot cycle whose members are all positive.

    {45840,51168} is the first example of a proper {2,1}-Aliquot cycle of order 2, and {65071776,77260656,82842816,89761152} which you found is the first example of a proper {2,1}-Aliquot cycle of order 4.

    Though, 10410218 {1,1}-Aliquot cycles of order 2 which is the same thing as Amicable pairs are known and  137 {1,1}-Aliquot cycle of order 4 are known.

    Where {1,1}-Aliquot sequence means ordinary  Aliquot sequence.            .   
    See Pedersen's site "Table of Aliquot cycles".

    http://amicable.homepage.dk/tables.htm

    I wonder if a {2,1}-Aliquot sequence which goes to infinity exists.
    I typed 44 digits numbers at random on key board and calculated the sequence such that a_{n}=S(a_{n-1}), at last it became a cycle which has 7 members                          .
    Is the following conjecture correct?
    "All {2,1}-Aliquot sequences become cycles." 

    [Definition of {k.l}-Aliquot sequence]
    Let S(n)=Sigma(n)/l-k*n
    a_{m}=S(a_{m-1})

    Yasutoshi



>Neil,
>
>	I fixed some typographical errors in the title and the keywords (moniker
>nonn), added the Mathematica coding, corrected a(7) from 4750 to 4756, and
>extended the sequence and %Comment line. I also changed the %Comment line from
>"Each cycle has some negative integers as members. ..." since a(17)-a(20) do
>not have any negative integers as members in their cycles.
>
>Thanx, Bob.	
>
>
>%I A113285
>%S A113285 51,72,120,132,672,2602,4756,10054,14884,45840,51168,116252,523776,
>%T A113285 906202,3003698,5271836,65071776,77260656,82842816,89761152,138357404,
>%U A113285 139626548
>%N A113285 Let S(n)=Sigma(n)-2n. Numbers such that S(S(S(S(n))))=n, {2,1}-Sociable number of order 1, 2 or 4.
>%C A113285 Each cycle may have some negative integers as members. If n is a negative integer then Sigma(n)=-Sigma(-n). Orders of each cycle are 4,4,1,4,1,4,4,4,4,2,2,2,1,2,4,4,4,4,4,4,4,4,..., .
>%H A113285 Eric Weisstein, <a href="http://mathworld.wolfram.com/SociableNumbers.html">WathWorld</a>
>%e A113285 {51,-30,132,72} is a {2,1}-Aliquot cycle.
>%t A113285 fQ[n_] := Nest[DivisorSigma[1, #] - 2# &, n, 4] == n; t = {}; Do[ If[fQ[n], AppendTo[t, n]], {n, 3*10^8}]; t (* RGWv *)
>%Y A113285 Cf. A113791.
>%K A113285 nonn,new
>%O A113285 1,1
>%A A113285 Yasutsohi Kohmoto zbi74583(AT)boat.zero.ad.jp, Jan 27 2006
>%E A113285 a(12)-a(22) from RGWv (rgwv(at)rgwv.com), Jan 30 2006
>
>
>