The happy function H : N → N sends a positive integer to the sum of the squares of its digits. A number x is said to be happy if the sequence {Hn(x)}∞ n=1 eventually reaches 1 (here Hn(x) denotes the nth iteration of H on x). A basic open question regarding happy numbers is what bounds on the density can be proved. This paper uses probabilistic methods to reduce this problem to experimentally finding suitably large intervals containing a high (or low) density of happy numbers as a subset. Specifically, we show that ˉ d > .18577 and d < .1138, where ˉ d and d denote the upper and lower density of happy numbers respectively. We also prove that the asymptotic density does not exist for several generalizations of happy numbers.
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