A happy number is defined by the following process. Starting with any positive integer, replace the number by the sum of the squares of its digits, and repeat the process until the number equals 1 (where it will stay), or it loops endlessly in a cycle which does not include 1. Those numbers for which this process ends in 1 are happy numbers, while those that do not end in 1 are unhappy numbers. A computer search up to 10²⁰ suggests that about 12% of numbers are happy^[1], though no proof is known (Guy 2004:§E34). A negative number is a number that is less than zero, such as −3. ... The integers are commonly denoted by the above symbol. ... Addition is one of the basic operations of arithmetic. ... In mathematics and computer science, a numerical digit is a symbol, e. ... A happy number is defined by the following process. ...

Overview

More formally, given a number n = n₀, define a sequence n₁, n₂, ... where n_{i + 1} is the sum of the squares of the digits of n_i. Then n is happy if and only if

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.

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.

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, 326, 329, 331, 338, 356, 362, 365, 367, 368, 376, 379, 383, 386, 391, 392, 397, 404, 409, 440, 446, 464, 469, 478, 487, 490, 496. (sequence A007770 in OEIS).

This article is about the number one. ... Seven Days of Creation - 1765 book, title page 7 (seven) is the natural number following 6 and preceding 8. ... This article is about the number 10. ... 13 (thirteen) is the natural number after 12 and before 14. ... 19 (nineteen) is the natural number following 18 and preceding 20. ... 23 (twenty-three) is the natural number following 22 and preceding 24. ... 28 (twenty-eight) is the natural number following 27 and preceding 29. ... 31 (thirty-one) is the natural number following 30 and preceding 32. ... 32 (thirty-two) is the natural number following 31 and preceding 33. ... 44 (forty-four) is the natural number following 43 and preceding 45. ... 49 (forty-nine) is the natural number following 48 and preceding 50. ... 68 (sixty-eight) is the natural number following 67 and preceding 69 // Sixty-eight is a nontotient. ... Look up seventy in Wiktionary, the free dictionary. ... 79 (seventy-nine) is the natural number following 78 and preceding 80. ... 82 is the natural number following 81 and preceding 83. ... 86 (eighty-six) is the natural number following 85 and preceding 87. ... 91 (ninety-one) is the natural number following 90 and preceding 92. ... 94 (ninety-four) is the natural number following 93 and preceding 95. ... 97 is the natural number following 96 and preceding 98. ... 100 (one hundred) (the Roman numeral is C for centum) is the natural number following 99 and preceding 101. ... 103 is the natural number following 102 and preceding 104. ... 109 is the natural number following 108 and preceding 110. ... 129 is the natural number following 128 and preceding 130. ... 130 is the natural number following 129 and preceding 131. ... 133 is the natural number following 132 and preceding 134. ... 139 (One hundred thirty-nine) is the natural number following 138 and preceding 140. ... 167 is the natural number following 166 and preceding 168. ... Cardinal one hundred and seventy-six Ordinal 176th (one hundred and seventy-sixth) Factorization Divisors 1, 2, 4, 8, 11, 16, 22, 44, 88, 176 Roman numeral CLXXVI Binary 10110000 Hexadecimal B0 176 is an even natural number between 175 and 177. ... Cardinal One hundred [and] eighty-eight Ordinal 188th Factorization Roman numeral CLXXXVIII Binary 10111100 Hexadecimal BC 188 is the natural number following 187 and preceding 189. ... 190 is the natural number following one hundred [and] eighty-nine and preceding one hundred [and] ninety-one. ... 190 is the natural number following one hundred eighty-nine and preceding one hundred ninety-one. ... 193 is the natural number between 192 and 194. ... 200 (two hundred) is the natural number following 199 and preceding 201. ... 200 (two hundred) is the natural number following 199 and preceding 201. ... 210 is the natural number following 209 and preceding 211. ... 220 (two hundred [and] twenty) is the natural number following 219 and preceding 221. ... 230 (two hundred [and] thirty) is the natural number following 229 and preceding 231. ... 230 (two hundred [and] thirty) is the natural number following 229 and preceding 231. ... 239 (two hundred [and] thirty-nine) is the natural number following 238 and preceding 240. ... 260 (two hundred [and] sixty) is the magic constant of the n×n normal magic square and n-Queens Problem for n = 8, the size of an actual chess board. ... 263 is the natural number between 262 and 264. ... 280 is the natural number after 279 and before 281. ... 290 is the natural number after 289 and before 291. ... 290 is the natural number after 289 and before 291. ... Three hundred is the natural number following two hundred ninety-nine and preceding three hundred one. ... This article is about the number 300. ... This article is about the number 300. ... 313 is the integer following 312 and preceding 314. ... This article is about the number 300. ... This article is about the number 300. ... This article is about the number 300. ... This article is about the number 300. ... This article is about the number 300. ... This article is about the number 300. ... This article is about the number 300. ... This article is about the number 300. ... Previous number: 364 Next number: 366 365 is a semiprime centered square number. ... 367 is John Galts apartment number in the book Atlas Shrugged, by Ayn Rand. ... This article is about the number 300. ... Four hundred and ninety-six is the natural number following four hundred and ninety-five and preceding four hundred and ninety-seven. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ...

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,

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. Exhaustive search then shows that every number in the interval [1,99] is either happy or goes to the above cycle. In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ...

Happy primes

A happy prime is a happy number that is prime. The first few happy primes are In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. ...

7, 13, 19, 23, 31, 79, 97, 103, 109, 139, 167, 193, 239, 263, 293, 313, 331, 367, 379, 383, 397, 409, 487 (sequence A035497 in OEIS) The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ...

Note that all primes of the form 10ⁿ + 3 and 10ⁿ + 9 are happy. The palindromic prime 10¹⁵⁰⁰⁰⁶ + 7426247×10⁷⁵⁰⁰⁰ + 1 is also a happy prime with 150007 digits because the many 0's do not contribute to the sum of squared digits, and 1² + 7² + 4² + 2² + 6² + 2² + 4² + 7² + 1² = 176, which is a happy number. Paul Jobling discovered the prime in 2005.^[2]. A palindromic prime is a prime number that is also a palindromic number. ...

As of June 2007, the largest known happy prime and the twelfth largest known prime is 4847 × 2^3321063 + 1. The decimal expansion has 999744 digits: 1844857508...(999724 digits omitted)...2886501377. Richard Hassler and Seventeen or Bust discovered the prime in 2005.^{[3] [4]} Jens K. Andersen identified it as the largest known happy prime in June 2007. June 2007 is the sixth month of that year. ... Seventeen or Bust is a distributed computing project to solve the last seventeen cases in the Sierpinski problem. ...

In the Doctor Who episode "42", a sequence of happy primes (313, 331, 367, 379) is used as a code for unlocking a sealed door on a spaceship falling into a star. This article is about the television series. ... 42 is an episode of the British science fiction television series Doctor Who. ... In communications, a code is a rule for converting a piece of information (for example, a letter, word, or phrase) into another form or representation, not necessarily of the same type. ...

Happy numbers in other bases

The definition of happy numbers depends on the decimal (i.e., base 10) representation of the numbers. The definition can be extended to other bases. In mathematics, the base or radix is the number of various unique symbols (digits), including zero, that a positional numeral system uses to represent numbers in a given counting system. ...

To represent numbers in other bases, we may use a subscript to the right to indicate the base. For instance, 100₂ represents the number 4, and

Then, it is easy to see that there are happy numbers in every base. For instance, the numbers

1_b,10_b,100_b,1000_b,...

are all happy, for any base b.

By a similar argument to the one above for decimal happy numbers, we can see that unhappy numbers in base b lead to cycles of numbers less than 1000_b. We can use the fact that if n < 1000_b, then the sum of the squares of the base-b digits of n is less than or equal to

3(b − 1)²

which can be shown to be less than b³. This shows that once the sequence reaches a number less than 1000_b, it stays below 1000_b, and hence must cycle or reach 1.

In base 2, all numbers are happy. All binary numbers larger than 1000₂ decay into a value equal to or less than 1000₂, and all such values are happy: The following four sequences contain all numbers less than 1000₂: The binary numeral system, or base-2 number system, is a numeral system that represents numeric values using two symbols, usually 0 and 1. ...

Since all sequences end in 1, we conclude that all numbers are happy in base 2. This makes base 2 a happy base.

The only known happy bases are 2 and 4, yet more could exist.

Origin

Happy numbers were brought to the attention of Reg Allenby[1], a British author and Senior Lecturer in pure mathematics at Leeds University, by his daughter. She had learned of them at school, but they "may have originated in Russia" (Guy 2004:§E34). Broadly speaking, pure mathematics is mathematics motivated entirely for reasons other than application. ... University Tower, University of Leeds The University of Leeds (United Kingdom) is amongst the largest of British universities and the most popular by applicants, with 52,444 applicants in 2003 for 7,228 places (UCAS). ... A happy number is defined by the following process. ...

Notes

1. ^ 12,005,034,444,292,997,294 happy numbers under 10²⁰ — http://www.research.att.com/~njas/sequences/A068571
2. ^ http://primes.utm.edu/primes/page.php?id=76550
3. ^ http://primes.utm.edu/primes/page.php?id=75994
4. ^ http://www.seventeenorbust.com/documents/prime-101205.txt

References

• Walter Schneider, Mathews: Happy Numbers.
• Eric W. Weisstein, Happy Number at MathWorld.
• Happy Numbers at The Math Forum.
• Guy, Richard (2004), Unsolved Problems in Number Theory (third edition), Springer-Verlag, ISBN 0-387-20860-7