On Weird and Pseudoperfect Numbers

Abstract

If n is a positive integer and a(n) denotes the sum of the divisors of n, then n is perfect if a(n) = 2n, abundant if 0(n) _ 2n and deficient if a(n) 2n and deficient if a(n) < 2n. We further define n to be pseudoperfect if n is the distinct sum of some of the proper divisors of n, e.g., 20 = 1 + 4 + 5 + 10 is pseudoperfect (6). An integer is called primitive abundant if it is abundant but all its proper divisors are deficient. It is primitive pseudoperfect if it is pseudoperfect but none of its proper divisors are pseudoperfect. An integer n is called weird if n is abundant but not pseudoperfect. The smallest weird number is 70 and Table 1 is a list of all weird numbers not exceeding 106. The study of weird numbers leads to surprising and unexpected difficulties. In particular, we could not decide whether there are any odd weird numbers (1) nor whether o-(n)/n could be arbitrarily large for weird n. We give an outline of the proof that the density of weird numbers is positive and discuss several related problems. Some of the proofs are only sketched, especially, if they are similar to proofs which are already in the literature. First, we consider the question of whether there are weird numbers n for which 0(n)/n can take on arbitrarily large values. Tentatively, we would like to suggest that the answer is negative. We can decide a few related questions. Let n be an integer with 1 = di < ... < dk = n the divisors of n. We say that n has property P if all the 2k sums k=l ,idi, i = 0 or 1, are distinct. P. Erdbs proved that the density