another aliquot

[y.kohmoto]

    Hello, sequfans.
    [another kind of aliquot sequence]
    The sequence which is defined as follows is called 1/m-sigma sequence :


    a(n)=1/m*sigma(a(n-1))


    Multiple Perfect Number is a fixed point of this mapping.
    Because,
    if a(n)=a(n-1) then
    m*a(n)=sigma(a(n))


    If the sequence becomes a cyclic sequence, then it is called a
1/m-sociable number of order k.
    k is number of the members.

    Three cases are possible :
    1. It becomes cyclic.
    2. It becomes divergent.
    3. It stops at n-th term which m doesn't divide sigma(a(n)).

    The longest record of 1/4-sosiable number.

    k=25
2 ^ 15 * 5 * 7 * 13 * 31 * 83 * 181
2 ^ 11 * 3 ^ 3 * 5 * 7 ^ 3 * 13 * 17 * 257
2 ^ 9 * 3 ^ 6 * 5 ^ 4 * 7 ^ 2 * 13 * 43
2 * 3 ^ 2 * 7 * 11 ^ 3* 19 * 31 * 71 * 1093
2 ^ 15 * 3 ^ 4 * 5 * 13 * 61 * 547
2 ^ 3 * 3 ^ 2 * 5 * 7 * 11 ^ 2 * 17 * 31 * 137* 257
2 ^ 10 * 3 ^ 6 * 5 * 7 * 13 * 19 * 23 * 43
2 ^ 10 * 3 ^ 2 * 5 * 7 * 11 * 23 * 89 * 1093
2 ^ 9 *3 ^ 5 * 5 * 13 * 23 * 89 * 547
2 ^ 8 * 3 ^ 5 * 5 * 7 ^ 2 * 11 * 13 * 31 * 137
2 ^ 10 * 3 ^ 4 * 7 ^ 3 * 13 * 19 * 23 * 73
2 ^ 9 * 3 * 5 ^ 3 * 7 * 11 ^ 2 * 23 * 37* 89
2 ^ 10 * 3 ^ 5 * 5 * 7 * 11 * 13 * 19 ^ 2 * 31
2 ^ 12 * 3 ^ 3 * 7 ^ 2 * 13 * 23 * 89 * 127
2 ^ 13 * 3 ^ 4 * 5 ^ 2 * 7 * 19 * 8191
2 ^ 16 * 3 * 5 * 11 ^ 2 * 31 * 43 * 127
2 ^ 15 * 3 * 7 * 11 * 19 * 131071
2 ^ 24* 3 ^ 2 * 5 ^ 2 * 17 * 257
3 ^ 3 * 13 * 31 ^ 2 * 43 * 601* 1801
2 ^ 6 * 3 * 5 * 7 ^ 2 * 11 * 17 * 43 * 53 * 331
2 ^ 9 * 3 ^ 8 * 11 * 19 * 83 * 127
2 ^ 11 * 3 ^ 3 * 5 * 7 * 11 * 13 * 31 * 757
2 ^ 14 * 3 ^ 4 * 5 ^ 2 * 7 ^ 2 * 13 * 379
2 * 3 * 5 * 7 ^ 2 * 11 ^ 2 * 19 ^ 2 * 31 ^ 2 * 151
2 ^ 4 * 3 ^ 5 * 7 * 19 ^ 3 * 127 * 331

    I know no mathematician who studies this sequence.
    Tell me any discovery about it.

    Yasutoshi
    http://mathworld.wolfram.com/SociableNumbers.html
    http://boat.zero.ad.jp/~zbi74583/another02.htm