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
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
907 | 1188 | 635 |
638 | 910 | 1182 |
1185 | 632 | 913 |
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