A happy number is a positive integer whose iterated sum of squared decimal digits
reaches 1. Let the sum of the squares
of the digits of a positive
integer 
be represented by 
.
In a similar way, let the sum of the squares
of the digits of 
be represented by 
, and so on.
Iteration from a positive integer eventually reaches 1 or enters the periodic
sequence 
(Porges 1945). If the starting value 0 is allowed, then 0 is an additional fixed
point. These ten periodic points form OEIS
A039943.
If 
for some 
, then the original integer 
is said to be happy. For example,
starting with 7 gives the sequence 7, 49, 97, 130, 10,
1, so 7 is a happy number.
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, ... (OEIS A007770). These are also the numbers whose 2-recurring digital invariant sequences have period 1. The numbers of iterations required for these to reach 1 are 0, 5, 1, 2, 4, 3, 3, 2, 3, 4, 4, 2, 5, ... These counts are one less than the corresponding terms of OEIS A090425, which counts the terms in each trajectory, including its starting value and the final 1.
The numbers of happy numbers less than or equal to 1, 
, 
, ... are given by 1, 3, 20, 143, 1442, 14377, 143071, ...
(OEIS A068571).
The first few consecutive happy numbers 
have 
, 129, 192, 262, 301, 319, 367, 391, ... (OEIS A035502).
Similarly, the first few happy triplets start with 1880, 4780, 4870, 7480, 7839,
... (OEIS A072494).
The first few happy primes are 7, 13, 19, 23, 31, 79, 97, 103, 109, 139, ... (OEIS A035497).
Once it is known whether a number is happy (or not), then any number in the sequence 
, 
, 
,
... will also be happy (or not). A number that is not happy is called unhappy.
Unhappy numbers have eventually
periodic sequences of 
which never reach 1.
Any permutation of the digits of an unhappy or happy number must also be unhappy or happy. This follows from the fact that addition is commutative.
