Happy Number Checker

For any n with d digits in base 10, the digit-square sum is at most 81d. For n>1000, 81d < n, so the sequence eventually drops below 1000. Below 1000, every unhappy number eventually enters the unique cycle 4→16→37→58→89→145→42→20→4.

The density of happy numbers is approximately 14.3%. The happy numbers have a natural density, meaning the proportion stabilizes as n→∞. In any block of 10 consecutive numbers starting with a multiple of 10, typically 1-2 are happy.

Primes that are also happy: 7, 13, 19, 23, 31, 79, 97, 103, 109, 139, ... There are infinitely many happy primes. They combine two interesting properties and are used in recreational math puzzles.