In number theory, a happy number is a number which eventually reaches 1 when replaced by the sum of the square of each digit. For instance, 13 is a happy number because and. On the other hand, 4 is not a happy number because the sequence starting with and eventually reaches, the number that started the sequence, and so the process continues in an infinite cycle without ever reaching 1. A number which is not happy is called sad or unhappy. More generally, a -happy number is a natural number in a given number base that eventually reaches 1 when iterated over the perfect digital invariant function for. The origin of happy numbers is not clear. Happy numbers were brought to the attention of Reg Allenby by his daughter, who had learned of them at school. However, they "may have originated in Russia".
Happy numbers and perfect digital invariants
Formally, let be a natural number. Given the perfect digital invariant function for base, a number is -happy if there exists a such that, where represents the -th iteration of, and -unhappy otherwise. If a number is a [|nontrivial perfect digital invariant] of, then it is -unhappy. For example, 19 is 10-happy, as For example, 347 is 6-happy, as There are infinitely many -happy numbers, as 1 is a -happy number, and for every, is -happy, since its sum is 1. Indeed, the happiness of a number is preserved by removing or inserting zeroes at will, since they do not contribute to the cross sum.
A happy base is a number base where every number is -happy. The only happy bases less than are base 2 and base 4.
Specific -happy numbers
4-happy numbers
For, the only positive perfect digital invariant for is the trivial perfect digital invariant 1, and there are no other cycles. Because all numbers are preperiodic points for, all numbers lead to 1 and are happy. As a result, base 4 is a happy base.
6-happy numbers
For, the only positive perfect digital invariant for is the trivial perfect digital invariant 1, and the only cycle is the eight-number cycle and because all numbers are preperiodic points for, all numbers either lead to 1 and are happy, or lead to the cycle and are unhappy. Because base 6 has no other perfect digital invariants except for 1, no positive integer other than 1 is the sum of the squares of its own digits. In base 10, the 74 6-happy numbers up to 1296 = 64 are:
10-happy numbers
For, the only positive perfect digital invariant for is the trivial perfect digital invariant 1, and the only cycle is the eight-number cycle and because all numbers are preperiodic points for, all numbers either lead to 1 and are happy, or lead to the cycle and are unhappy. Because base 10 has no other perfect digital invariants except for 1, no positive integer other than 1 is the sum of the squares of its own digits. In base 10, the 143 10-happy numbers up to 1000 are: The distinct combinations of digits that form 10-happy numbers below 1000 are : The first pair of consecutive 10-happy numbers is 31 and 32. The first set of three consecutive is 1880, 1881, and 1882. It has been proven that there existsequences of consecutive happy numbers of any natural-number length. The beginning of the first run of at least n consecutive 10-happy numbers for n = 1, 2, 3,... is The number of 10-happy numbers up to 10n for 1 ≤n ≤ 20 is
Happy primes
A -happy prime is a number that is both -happy and prime. Unlike happy numbers, rearranging the digits of a -happy prime will not necessarily create another happy prime. For instance, while 19 is a 10-happy prime, 91 = 13 × 7 is not prime. All prime numbers are 2-happy and 4-happy primes, as base 2 and base 4 are happy bases.
6-happy primes
In base 6, the 6-happy primes below 1296 = 64 are
10-happy primes
In base 10, the 10-happy primes below 500 are The palindromic prime is a 10-happy prime with digits because the many 0s do not contribute to the sum of squared digits, and = 176, which is a 10-happy number. Paul Jobling discovered the prime in 2005. , the largest known 10-happy prime is 242643801 − 1. Its decimal expansion has digits.
Programming example
The examples below implements the perfect digital invariant function for and a default base described in the definition of happy given at the top of this article, repeatedly; after each time, they check for both halt conditions: reaching 1, and repeating a number. A simple test in Python to check if a number is happy: def pdi_function: """Perfect digital invariant function.""" total = 0 while number > 0: total = total + pow number = number // base return total def is_happy -> bool: """Determine if the specified number is a happy number.""" seen_numbers = while number > 1 and number not in seen_numbers: seen_numbers.append number = pdi_function return number 1
