happy number

Every natural number is either 'happy' or 'unhappy' (a trivial name for an interesting property), determined by the following test:

Let d_k, d_k-1, ... , d₂, d₁, d₀ be the decimal digits of a number n=s₀. Let s₁ = d_k² + d_k-1² + ... + d₁² + d₀², the sum of the squares of the digits of s₀. Similarly, let s₂ be the sum of the squares of the digits of s₁, and so on. If there exists an i>1 such that s_i=1, then n=s₀, s₁, ..., and s_i-1 are said to be happy numbers. Otherwise, for all i>0, s_i is said to be an unhappy number. Further, all permutations on the digits of s_i are just as happy or unhappy as s_i, since addition is commutative.

For example, 28 is happy because 2²+8²=4+64=68, 6²+8²=36+64=100, and 1²+0²+0²=1+0+0=1. The first several happy numbers are 1, 7, 10, 13, 19, 23, 28, ... It is unfortunate that my favorite number, 4, is unhappy. =(

So far, the 'happiness' property of a number is merely interesting, relating only to digital invariance.

Information gleaned from http://mathworld.wolfram.com/HappyNumber.html.