The aliquot sequence starting from n is defined as follows: let σ(n) be the sum of divisors of n, then one simply computes f(n)=σ(n)-n, and one iterates. For example, if we start from 12, whose divisors are 1, 2, 3, 4, 6, 12, then σ(12)=1+2+3+4+6+12=28 and therefore f(12)=16. Then f(16)=15, f(15)=9, f(9)=4, f(4)=3, f(3)=1, f(1)=0, f(0)=0, and one then loops on 0. One can also loop on perfect numbers, i.e. numbers such that f(n)=n, for example n=6. An open question asked by M. E. Catalan in 1888 is whether any aliquot sequence eventually reaches 1, a perfect number, or a cycle of amicable or sociable numbers. Lehmer tried to investigate with the computer the sequences of starting value less than 1000, and found that all terminate, except perhaps for n=276, 552, 564, 660 and 966.
Continuing work of Wolfgang Creyaufmueller, with the help
of Sam Wagstaff, Arjen Lenstra, and Peter Montgomery,
I have extended
the ``Lehmer five'' sequences, together with the sequences starting from
1074, 1134, 19560 and 204828 (sequence
is also interesting):
|starting value||current index|