Let us define a function  , for  0$">, which gives the sum of the squares of the digits of  , so, for example,  .

If we start from a number    and we repeatedly apply  , we obtain a sequence    of numbers  ,  ,  , and so on.

A number    is called happy if    contains the number 1.

Note that  , so in that case the sequence    has an infinite tail of  's.

If a number is not happy then it is easy to see that at a certain point    will enter the infinite loop

So, for example, starting from    we obtain  , so    is happy. See figure aside.

On the contrary, starting from 61 we obtain    and thus 61 is not happy, since 89 belongs to the unhappy loop.

The first  -tuple of consecutive happy numbers, for    starts at 31, 1880, 7839, and 44488, respectively.

E. El-Sedy & S. Siksek proved that there can be runs of arbitrary length.

There are 3, 20, 143, 1442, 14377, 143071,... happy numbers up to 10, 100, 1000,....

The smallest 3 × 3 magic square whose entries are happy numbers is

The first 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 more terms