Happy number

Consider the following process: given a number, take the sum of the squares of its digits. A number is a happy number if iterating this process eventually yields 1. Numbers that are not happy are called unhappy numbers.

More formally, given a number $n=n_{0}$, define a sequence $n_{1}$, $n_{2}$, ... where $n_{i+1}$ is the sum of the squares of the digits of $n_{i}$. Then $n$ is happy if and only if this sequence goes to 1.

For example, 7 is happy, as the associated sequence is:

7² = 49

4² + 9² = 97

9² + 7² = 130

1² + 3² + 0² = 10

1² + 0² = 1

If a number is happy, then all members of its sequence are happy; if a number is unhappy, all members of its sequence are unhappy.

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, 103, 109, 129, 130, 133, 139, 167, 176, 188, 190, 192, 193, 203, 208, 219, 226, 230, 236, 239, 262, 263, 280, 291, 293, 301, 302, 310, 313, 319, 320. (sequence A007770 in the OEIS)

A computer search up to 10¹⁰ suggests that about 15 percent of numbers are happy, though no proof is known.

Sequence behavior

If $n$ is not happy, then its sequence does not go to 1. What happens instead is that it ends up in the cycle

4, 16, 37, 58, 89, 145, 42, 20, 4, ...

To see this fact, first note that if $n$ has $m$ digits, then the sum of the squares of its digits is at most $81m$. For $m=4$ and above,

$n\geq 10^{m-1}>81m$

so any number over 1000 gets smaller under this process. Once we are under 1000, the number for which the sum of squares of digits is largest is 999, and the result is 3 times 81, that is, 243.

• In the range 100 to 243, the number 199 produces the largest next value, of 163.
• In the range 100 to 163, the number 159 produces the largest next value, of 107.
• In the range 100 to 107, the number 107 produces the largest next value, of 50.

Considering more precisely the intervals [244,999], [164,243], [108,163] and [100,107], we see that every number above 99 gets strictly smaller under this process. Thus, no matter what number we start with, we eventually drop below 100. A computer program can easily verify that in the range 1 to 99, every number is either happy, or goes to the above cycle.

Happy numbers in other bases

Sets of happy and unhappy numbers exist in every base. These all have similar behavior, such as happy numbers eventually iterating to 1, and all unhappy number sequences leading to infinite loops consisting of values less than 1000_b, where b is an accepted sub script for a base.

Similar to the above sequence behavior, any number less than 1000_b has the value $B^{3}-1$ and can't have a digit sum any larger than $3*(B-1)^{2}$, or $3B^{2}-6B+3$. Since this is true for all $B>0$ any acceptable base will also have repeating unhappy number sequences less than 1000_b.

Binary, also known as Base 2, only has happy numbers. All binary numbers larger than 1000₂ decay into a value equal to or less than 1000₂, and all such values are happy. Therefore, all numbers in base two are considered happy. This makes base 2 a Happy Base.

Each of the following are in base 2:

{111} -> 11

{11, 101, 110} -> 10

{1, 10, 100, 1000} -> 1

The only known Happy bases are 2 and 4, yet more could exist. Although, it is impossible to have an Unhappy base unless the base itself was infinite, many bases don't even have happy numbers between one and their value 10_b. Here are some of the bases without this common property:

(9, 10, 16, 18, 19, 20, 29, 30, 34, ...)

References

• Walter Schneider, Mathews: Happy Numbers.
• Weisstein, Eric W. "Happy Number". MathWorld.
• Happy Numbers at The Math Forum.

External links

• Happy number calculator - online arbitrary-base happy number calculator]