Consider the following process: given a number, take the sum of the squares of its digits. A number is a happy number if iterating this process eventually yields 1. Numbers that are not happy are called unhappy numbers.

More formally, given a number n = n₀, define a sequence n₁, n₂, ... where n_{i + 1} is the sum of the squares of the digits of n_i. Then n is happy if and only if this sequence goes to 1.

For example, 7 is happy, as the associated sequence is:

7² = 49

4² + 9² = 97

9² + 7² = 130

1² + 3² + 0² = 10

1² + 0² = 1

If a number is happy, then all members of its sequence are happy; if a number is unhappy, all members of its sequence are unhappy.

The first few happy numbers are

1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97, 100, 103, 109, 129, 130, 133, 139 , 167, 176 , 188 , 190, 192 , 193 , 203 , 208 , 219 , 226 , 230 , 236 , 239, 262, 263, 280, 291, 293, 301, 302, 310, 313, 319, 320.

Computer search up to 10¹⁰ suggests that about 15 percent of numbers are happy, though no proof is known.

Sequence behavior

If n is not happy, then its sequence does not go to 1. What happens instead is that it ends up in the cycle

4, 16, 37, 58, 89, 145, 42, 20, 4, ...

To see this fact, first note that if n has m digits, then the sum of the squares of its digits is at most 81m. If m > 3 then

so any number over 1000 gets smaller under this process. Once we are under 1000, the number for which the sum of squares of digits is largest is 999, and the result is 3 times 81, that is, 243.

• In the range 100 to 243, the number 199 produces the largest next value, of 163.
• In the range 100 to 163, the number 159 produces the largest next value, of 107.
• In the range 100 to 107, the number 107 produces the largest next value, of 50.

We see that every number above 99 gets smaller under this process. Thus, no matter what number you start with, you eventually drop below 100. A computer program can easily verify that in the range 1 to 99, every number is either happy, or goes to the above cycle.

References

• Walter Schneider, Mathews: Happy Numbers.
• Happy Numbers at The Math Forum.