b6
aliquot sequences
Let n be a positive integer. We denote by s(n) the sum of all positive divisors of n other than n. We call s the aliquot sum function. The aliquot sequence {sk(n)} (k=0,1,2..) is the sequence obtained by iterating the function s.
For example, when n=220, s2k(n)=220, s2k+1(n)=284 hence it is a periodic sequence. When n=138, after reaching a maximum s117(138)=179931895322, it terminates at s177(138)=1.
It is unknown whether there exists any integer n such that the aliquot sequence is not bounded, i.e. sk(n)→∞ (k→∞) .
The smallest unknown n is 276 .
long aliquot sequences
terminates or reaches to an aliquot cycle
unknown
external links
Aliquot Sequences (Paul Zimmermann)
Aliquot Sequences (Wolfgang Creyaufmueller)