Amicable m-ple

[koh]

    Hi, Seqfans


    I considered about sequences of the smallest members of m-multiple Perfect Number and Amicable m-ple and 1/2-Amicable 2*m-ple. 

    The definitions are as follows.

    m-PN :     Sigma(n)=m*n
    A m-ple :    Sigma(n_i)=Sum_{1<=i<=m} n_i , 1<=i<=m
    1/2-A 2*m-ple : Sigma(n_i)=1/2*(Sum_{1<=i<=2*m} n_i) ,1<=i<=2*m

    If for all i n_i = n  then these definitions becomes the same, so I supposed that the sizes of m-PN and A m-ple and 1/2-A 2*mple are almost the same.
    But it doesn't seem to be correct.                   .
    Between their sizes, some relationship might exist.

    
    The smallest examples of m-PN and A m-ple and 1/2-A 2*m-ple are as follows.

    m               PN                                    A m-ple                        1/2-A 2*m-ple 
    1                1                                          1                                   no
    2                2                                        220                            32 digits  kn
    3              120                                       1980                    60681331968409600  kn                 
    4            30240                                    3270960                   174547933583278080  kn
    5      14182439040                          22749116458844160 kn                         un 
    6               21 digits                               un                               un
    7               57 digits                               un                               un
    8              133 digits                               un                               un
    9              287 digits                               un                               un
   10              685 digits                               un                               un
   11             1907 digits                               un                               un
                                  Where "un" means "unknown", "kn" means "known the smallest".
    
    %S A000001 1,2,120,30240,14182439040
    %N A000001 The smallest m-multiple Perfect Number.


    %S A000002 1,220,1980,3270960, 22749116458844160
    %N A000002 The smallest member of the smallest Amicable m-ple.
    %C A000002 There is no proof that (227491164588441600, 228507506351308800, 229862628701798400, 230878970464665600,243752632794316800) is the smallest Amicable Quintuple.
                  

    %S A000003 5702524371272605100551486698903,60681331968409600, 174547933583278080
    %N A000003 The smallest member of the smallest 1/2-Amicable 2*m-ple.
    %C A000003 The first term must be much smaller than this example.


    A000001 already exists on OEIS.
    I will submit A000002.
    I am not sure if A000003 fits to OEIS.

    Yasutoshi