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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 .^{[1]}

The origin of happy numbers is not clear. Happy numbers were brought to the attention of Reg Allenby (a British author and senior lecturer in pure mathematics at Leeds University) by his daughter, who had learned of them at school. However, they "may have originated in Russia" (Guy 2004:§E34).

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 , ( in base ) is -happy, since its sum is 1. The *happiness* of a number is preserved by removing or inserting zeroes at will, since they do not contribute to the cross sum.

By inspection of the first million or so 10-happy numbers, it appears that they have a natural density of around 0.15. Perhaps surprisingly, then, the 10-happy numbers do not have an asymptotic density. The upper density of the happy numbers is greater than 0.18577, and the lower density is less than 0.1138.^{[2]}

A happy base is a number base where every number is -happy. The only happy integer bases less than 5×10^{8} are base 2 and base 4.^{[3]}

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.

For , the only positive perfect digital invariant for is the trivial perfect digital invariant 1, and the only cycle is the eight-number cycle

- 5 → 41 → 25 → 45 → 105 → 42 → 32 → 21 → 5 → ...

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 = 6^{4} are (written in base 10):

- 1, 6, 36, 44, 49, 79, 100, 160, 170, 216, 224, 229, 254, 264, 275, 285, 289, 294, 335, 347, 355, 357, 388, 405, 415, 417, 439, 460, 469, 474, 533, 538, 580, 593, 600, 608, 628, 638, 647, 695, 707, 715, 717, 767, 777, 787, 835, 837, 847, 880, 890, 928, 940, 953, 960, 968, 1010, 1018, 1020, 1033, 1058, 1125, 1135, 1137, 1168, 1178, 1187, 1195, 1197, 1207, 1238, 1277, 1292, 1295

For , the only positive perfect digital invariant for is the trivial perfect digital invariant 1, and the only cycle is the eight-number cycle

- 4 → 16 → 37 → 58 → 89 → 145 → 42 → 20 → 4 → ...

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:

- 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, 536, 556, 563, 565, 566, 608, 617, 622, 623, 632, 635, 637, 638, 644, 649, 653, 655, 656, 665, 671, 673, 680, 683, 694, 700, 709, 716, 736, 739, 748, 761, 763, 784, 790, 793, 802, 806, 818, 820, 833, 836, 847, 860, 863, 874, 881, 888, 899, 901, 904, 907, 910, 912, 913, 921, 923, 931, 932, 937, 940, 946, 964, 970, 973, 989, 998, 1000 (sequence A007770 in the OEIS).

The distinct combinations of digits that form 10-happy numbers below 1000 are (the rest are just rearrangements and/or insertions of zero digits):

- 1, 7, 13, 19, 23, 28, 44, 49, 68, 79, 129, 133, 139, 167, 188, 226, 236, 239, 338, 356, 367, 368, 379, 446, 469, 478, 556, 566, 888, 899. (sequence A124095 in the OEIS).

The first pair of consecutive 10-happy numbers is 31 and 32.^{[4]} The first set of three consecutive is 1880, 1881, and 1882.^{[5]} It has been proven that there exist sequences of consecutive happy numbers of any natural number length.^{[6]} The beginning of the first run of at least *n* consecutive 10-happy numbers for *n* = 1, 2, 3, ... is^{[7]}

- 1, 31, 1880, 7839, 44488, 7899999999999959999999996, 7899999999999959999999996, ...

As Robert Styer puts it in his paper calculating this series: "Amazingly, the same value of N that begins the least sequence of six consecutive happy numbers also begins the least sequence of seven consecutive happy numbers."^{[8]}

The number of 10-happy numbers up to 10^{n} for 1 ≤ *n* ≤ 20 is^{[9]}

- 3, 20, 143, 1442, 14377, 143071, 1418854, 14255667, 145674808, 1492609148, 15091199357, 149121303586, 1443278000870, 13770853279685, 130660965862333, 1245219117260664, 12024696404768025, 118226055080025491, 1183229962059381238, 12005034444292997294.

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 (but is still 10-happy).

All prime numbers are 2-happy and 4-happy primes, as base 2 and base 4 are happy bases.

In base 6, the 6-happy primes below 1296 = 6^{4} are

- 211, 1021, 1335, 2011, 2425, 2555, 3351, 4225, 4441, 5255, 5525

In base 10, the 10-happy primes below 500 are

- 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 the OEIS).

The palindromic prime 10^{150006} + 7426247×10^{75000} + 1 is a 10-happy prime with 150007 digits because the many 0s do not contribute to the sum of squared digits, and 1^{2} + 7^{2} + 4^{2} + 2^{2} + 6^{2} + 2^{2} + 4^{2} + 7^{2} + 1^{2} = 176, which is a 10-happy number. Paul Jobling discovered the prime in 2005.^{[10]}

As of 2010^{[update]}, the largest known 10-happy prime is 2^{42643801} − 1 (a Mersenne prime).^{[dubious – discuss]} Its decimal expansion has 12837064 digits.^{[11]}

In base 12, there are no 12-happy primes less than 10000, the first 12-happy primes are (the letters X and E represent the decimal numbers 10 and 11 respectively)

- 11031, 1233E, 13011, 1332E, 16377, 17367, 17637, 22E8E, 2331E, 233E1, 23955, 25935, 25X8E, 28X5E, 28XE5, 2X8E5, 2E82E, 2E8X5, 31011, 31101, 3123E, 3132E, 31677, 33E21, 35295, 35567, 35765, 35925, 36557, 37167, 37671, 39525, 4878E, 4X7X7, 53567, 55367, 55637, 56357, 57635, 58XX5, 5X82E, 5XX85, 606EE, 63575, 63771, 66E0E, 67317, 67371, 67535, 6E60E, 71367, 71637, 73167, 76137, 7XX47, 82XE5, 82EX5, 8487E, 848E7, 84E87, 8874E, 8X1X7, 8X25E, 8X2E5, 8X5X5, 8XX17, 8XX71, 8E2X5, 8E847, 92355, 93255, 93525, 95235, X1X87, X258E, X285E, X2E85, X85X5, X8X17, XX477, XX585, E228E, E606E, E822E, EX825, ...

The examples below implement 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(number, base: int = 10):
"""Perfect digital invariant function."""
total = 0
while number > 0:
total += pow(number % base, 2)
number = number // base
return total
def is_happy(number: int) -> bool:
"""Determine if the specified number is a happy number."""
seen_numbers = set()
while number > 1 and number not in seen_numbers:
seen_numbers.add(number)
number = pdi_function(number)
return number == 1
```

- In 2007, the concept of happy numbers was used in
*Professor Layton and the Diabolical Box*, in puzzle 149 ("Number Cycle"), using the sequence beginning with 4, which repeats every 8 terms.

**^**"Sad Number". Wolfram Research, Inc. Retrieved 16 September 2009.**^**Gilmer, Justin (2013). "On the Density of Happy Numbers".*Integers*.**13**(2): 2. arXiv:1110.3836. Bibcode:2011arXiv1110.3836G.**^**Sloane, N. J. A. (ed.). "Sequence A161872 (Smallest unhappy number in base n)".*The On-Line Encyclopedia of Integer Sequences*. OEIS Foundation.**^**Sloane, N. J. A. (ed.). "Sequence A035502 (Lower of pair of consecutive happy numbers)".*The On-Line Encyclopedia of Integer Sequences*. OEIS Foundation. Retrieved 8 April 2011.**^**Sloane, N. J. A. (ed.). "Sequence A072494 (First of triples of consecutive happy numbers)".*The On-Line Encyclopedia of Integer Sequences*. OEIS Foundation. Retrieved 8 April 2011.**^**Pan, Hao (2006). "Consecutive Happy Numbers". arXiv:math/0607213.**^**Sloane, N. J. A. (ed.). "Sequence A055629 (Beginning of first run of at least*n*consecutive happy numbers)".*The On-Line Encyclopedia of Integer Sequences*. OEIS Foundation.**^**Styer, Robert (2010). "Smallest Examples of Strings of Consecutive Happy Numbers".*Journal of Integer Sequences*.**13**: 5. 10.6.3 – via University of Waterloo. Cited in Sloane "A055629".**^**Sloane, N. J. A. (ed.). "Sequence A068571 (Number of happy numbers <= 10^n)".*The On-Line Encyclopedia of Integer Sequences*. OEIS Foundation.**^**Chris K. Caldwell. "The Prime Database: 10^{150006}+ 7426247 · 10^{75000}+ 1".*utm.edu*.**^**Chris K. Caldwell. "The Prime Database: 2^{42643801}− 1".*utm.edu*.

- Guy, Richard (2004).
*Unsolved Problems in Number Theory*(3rd ed.). Springer-Verlag. ISBN 0-387-20860-7.

- Schneider, Walter: Mathews: Happy Numbers.
- Weisstein, Eric W. "Happy Number".
*MathWorld*. - calculate if a number is happy Archived 25 January 2019 at the Wayback Machine
- Happy Numbers at The Math Forum.
- 145 and the Melancoil at Numberphile.
- Symonds, Ria. "7 and Happy Numbers".
*Numberphile*. Brady Haran. Archived from the original on 15 January 2018. Retrieved 2 April 2013.