ALIQUOT SEQUENCE 3630 ENDS AFTER REACHING 100 DIGITS

Abstract

Abstract. In this paper we present a new computational record: the aliquot sequence starting at 3630 converges to 1 after reaching a hundred decimal digits. Also, weshow thecurrent status of all thealiquot sequences starting with a number under 10000; we have reached at leat 95 digits for all of them. In particular, we have reached at least 112 digits for the so-called “Lehmer five sequences”, and 101 digits for the “Godwin twelve sequences”. Finally, we give a summary showing the number of aliquot sequences of unknown end starting with a number ≤ 10 6. For a positive integer n, letσ(n) denote the sum of its divisors (including 1 and n), and s(n) =σ(n) − n the sum of its proper divisors (without n). A perfect number is a number n such that s(n) =n, and an amicable pair of numbers (n, m) satisfies s(n) =m, s(m) =n. In a similar way, tuples of numbers (a1,a2,...,al) such that s(ai) =ai+1 for 1 ≤ i ≤ l − 1ands(al) =a1 are known as aliquot cycles or sociable numbers. Given n, the way to compute σ(n) (and then, s(n)) is as follows. We find the prime decomposition of n = p a1 1 ···pad d.Then (1) σ(p a