# another aliquot

Wed Apr 9 10:08:00 CEST 2003

```    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