The Art and Craft of Problem Solving

9.4 POWER SERIES AND EULERIAN MATHEMATICS 355

9.4.37 Analogous to the concept of perfect numbers
(see Problems 7.5.33-7.5.35) are the abundant num­
bers. The natural number n is considered abundant if
a(n) > 2n.
(a) How abundant can a number get? In other
words. what is the largest possible value for the
ratio a(n)/n?

(b) What is the expected value of this "abundancy

quotient" a(n)/n? In other words. if you pick

an integer n at random. and compute the value

of a(n)/n. what limiting average value do we

get if we repeat this experiment indefinitely?

(c) What relative fraction of positive integers is

abundant?