Aspiring number

In mathematics, an aliquot sequence is a sequence of positive integers in which each term is the sum of the proper divisors of the previous term. If the sequence reaches the number 1, it ends, since the sum for 1 is 0. The aliquot sequence starting with a positive integer k can be defined formally in terms of the sum-of-divisors function σ₁ or the aliquot sum function s in the following way:

and s(0) is undefined.

For example, the aliquot sequence of 10 is 10, 8, 7, 1, 0 because:

Many aliquot sequences terminate at zero; all such sequences necessarily end with a prime number followed by 1 (since the only proper divisor of a prime is 1), followed by 0 (since 1 has no proper divisors). See (sequence in the OEIS) for a list of such numbers up to 75. There are a variety of ways in which an aliquot sequence might not terminate:

The lengths of the Aliquot sequences that start at n are

The final terms (excluding 1) of the Aliquot sequences that start at n are

Numbers whose Aliquot sequence terminates in 1 are

Numbers whose Aliquot sequence terminates in a perfect number are

Numbers whose Aliquot sequence terminates in a cycle with length at least 2 are

Numbers whose Aliquot sequence is not known to be finite or eventually periodic are

An important conjecture due to Catalan is that every aliquot sequence ends in one of the above ways: with a prime number, a perfect number, or a set of amicable or sociable numbers. The alternative would be that a number exists whose aliquot sequence is infinite yet never repeats. Any one of the many numbers whose aliquot sequences have not been fully determined might be such a number. The first five candidate numbers are often called the Lehmer five (named after D.H. Lehmer): 276, 552, 564, 660, and 966.

As of April 2015^[update], there were 898 positive integers less than 100,000 whose aliquot sequences have not been fully determined, and 9190 such integers less than 1,000,000.

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