It also happens to be the largest happy number under 500.

Then, it is easy to see that there are happy numbers in every base.

An interesting question is to wonder about the density of happy numbers.

A 'happy number' is defined by the following process.

And the happy numbers have made any Republican who lives in a governor's mansion think he deserves a promotion to vice president.

The origin of happy numbers is not clear.

It is also a happy number.

By similar reasoning, all happy numbers of this type must have a remainder of 1 when dividing by 3.

It is a happy number, meaning that repeatedly summing the squares of its digits eventually leads to 1:

It is also a Fibonacci number, a happy number, and one of only 3 known Wilson primes.

Mixed with all the happy numbers that reflect Ruocchio's achievements, the day that changed his life has never left the front of his mind.

But I have had a number for, honestly, 12 years that's my happy number where I feel good about myself and my energy's high.

There are infinitely many happy numbers and infinitely many unhappy numbers.

When the Doctor learns that nobody on the spaceship besides himself has heard of happy numbers, he asks, "Don't they teach recreational mathematics anymore?"

The definition of happy numbers depends on the decimal (i.e., base 10) representation of the numbers.

By a similar argument to the one above for decimal happy numbers, unhappy numbers in base b lead to cycles of numbers less than .

The unique combinations of the happy numbers below 1,000 follow (the rest are just rearrangements and/or insertions of zero digits):

Happy Numbers at The Math Forum.

Walter Schneider, Mathews: Happy Numbers.

Twenty-eight is a harmonic divisor number, a happy number, a triangular number, a hexagonal number,and a centered nonagonal number.

Those numbers for which this process ends in 1 are happy numbers, while those that do not end in 1 are unhappy numbers (or sad numbers).

Childlike tales and happy numbers like Mississippi John Hurt's "Candy Man," give way to cautionary horror stories related in the same affable tone as the lighter vignettes.

An interesting extension to the Happy Numbers problem is to find the sum of the cubes of the digits rather than the sum of the squares of the digits.

By conducting an exhaustive search of [1,2916] one finds that for summing the cubes of digits base 10 there are happy numbers and eight different types of unhappy number:

Happy numbers were brought to the attention of Reg Allenby (a British author and Senior Lecturer in pure mathematics at Leeds University) by his daughter, who had learned of them at school.