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{{short description|Numbers with a certain property involving recursive summation}}
{{distinguish|text=[[Harshad number]] (derived from Sanskrit ''harśa'' meaning "great joy")}}
{{Use dmy dates|date=October 2020}}
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 <math>1^2+3^2=10</math>, and <math>1^2+0^2=1</math>. On the other hand, 4 is not a happy number because the sequence starting with <math>4^2=16</math> and <math>1^2+6^2=37</math> eventually reaches <math>2^2+0^2=4</math>, 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 <math>b</math>-'''happy number''' is a [[natural number]] in a given [[number base]] <math>b</math> that eventually reaches 1 when iterated over the [[Perfect digital invariant|perfect digital invariant function]] for <math>p = 2</math>.<ref>{{cite web|url=http://mathworld.wolfram.com/SadNumber.html|title=Sad Number|publisher=Wolfram Research, Inc.|access-date=2009-09-16}}</ref>

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" {{harvcol|Guy|2004|p=§E34}}.

== Happy numbers and perfect digital invariants ==
{{See also|Perfect digital invariant}}
Formally, let <math>n</math> be a natural number. Given the [[Perfect digital invariant|perfect digital invariant function]] 
:<math>F_{p, b}(n) = \sum_{i=0}^{\lfloor \log_{b}{n} \rfloor} {\left(\frac{n \bmod{b^{i+1}} - n \bmod{b^{i}}}{b^{i}}\right)}^p</math>.
for base <math>b > 1</math>, a number <math>n</math> is <math>b</math>-happy if there exists a <math>j</math> such that <math>F_{2, b}^j(n) = 1</math>, where <math>F_{2, b}^j</math> represents the <math>j</math>-th [[Iterated function|iteration]] of <math>F_{2, b}</math>, and <math>b</math>-unhappy otherwise. If a number is a [[#Perfect digital invariant|nontrivial perfect digital invariant]] of <math>F_{2, b}</math>, then it is <math>b</math>-unhappy.

For example, 19 is 10-happy, as
: <math>F_{2, 10}(19) = 1^2 + 9^2 = 82</math>
: <math>F_{2, 10}^2(19) = F_{2, 10}(82) = 8^2 + 2^2 = 68</math>
: <math>F_{2, 10}^3(19) = F_{2, 10}(68) = 6^2 + 8^2 = 100</math>
: <math>F_{2, 10}^4(19) = F_{2, 10}(100) = 1^2 + 0^2 + 0^2 = 1</math>

For example, 347 is 6-happy, as
: <math>F_{2, 6}(347) = F_{2, 6}(1335_6) = 1^2 + 3^2 + 3^2 + 5^2 = 44</math>
: <math>F_{2, 6}^2(347) = F_{2, 6}(44) = F_{2, 6}(112_6) = 1^2 + 1^2 + 2^2 = 6</math>
: <math>F_{2, 6}^3(347) = F_{2, 6}(6) = F_{2, 6}(10_6) = 1^2 + 0^2 = 1</math>

There are infinitely many <math>b</math>-happy numbers, as 1 is a <math>b</math>-happy number, and for every <math>n</math>, <math>b^n</math> (<math>10^n</math> in base <math>b</math>) is <math>b</math>-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.

=== Natural density of ''b''-happy numbers ===
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.<ref>{{Cite journal |title=On the Density of Happy Numbers |journal=Integers |volume=13 |issue=2 |arxiv=1110.3836 |last1=Gilmer |first1=Justin |year=2013 |page=2 |bibcode=2011arXiv1110.3836G}}</ref>

=== Happy bases ===
{{unsolved|mathematics|Are [[base 2]] and [[base 4]] the only bases that are happy?}}
A happy base is a number base <math>b</math> where every number is <math>b</math>-happy. The only happy bases less than {{val|5e8}} are [[base 2]] and [[base 4]].<ref>{{Cite OEIS|sequencenumber=A161872|name=Smallest unhappy number in base n}}</ref>

==Specific ''b''-happy numbers==
===4-happy numbers===
For <math>b = 4</math>, the only positive perfect digital invariant for <math>F_{2, b}</math> is the trivial perfect digital invariant 1, and there are no other cycles. Because all numbers are [[Periodic point#Iterated functions|preperiodic points]] for <math>F_{2, b}</math>, all numbers lead to 1 and are happy. As a result, [[base 4]] is a happy base.

===6-happy numbers===
For <math>b = 6</math>, the only positive perfect digital invariant for <math>F_{2, b}</math> 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 <math>F_{2, b}</math>, 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<sup>4</sup> 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

0
For <math>b = 10</math>, the only positive perfect digital invariant for <math>F_{2, b}</math> is the trivial perfect digital invariant 1, and the
and because all numbers are preperiodic points for <math>F_{2, b}</math>, 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.
1234567891011121314151617181920
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 {{OEIS|id=A007770}}.
2122232425262728293031323334353637383940
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. {{OEIS|id=A124095}}.
41424344454647484950515253545657585960
The first pair of consecutive 10-happy numbers is 31 and 32.<ref>{{cite OEIS |1=A035502 |2=Lower of pair of consecutive happy numbers |access-date=8 April 2011}}</ref> The first set of three consecutive is 1880, 1881, and 1882.<ref>{{cite OEIS |1=A072494 |2=First of triples of consecutive happy numbers |access-date=8 April 2011}}</ref> It has been proven that there exist sequences of consecutive happy numbers of any natural number length.<ref>{{Cite arXiv |title=Consecutive Happy Numbers |eprint=math/0607213 |last1=Pan |first1=Hao |year=2006}}</ref> The beginning of the first run of at least ''n'' consecutive 10-happy numbers for ''n''&nbsp;=&thinsp;1, 2, 3, ... is<ref name="Sloane-A055629">{{Cite OEIS |1=A055629 |2=Beginning of first run of at least ''n'' consecutive happy numbers}}</ref>
: 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."<ref>{{cite journal |last=Styer |first=Robert |year=2010 |page=5 |title=Smallest Examples of Strings of Consecutive Happy Numbers |journal=[[Journal of Integer Sequences]] |volume=13 |id=10.6.3 |url=https://cs.uwaterloo.ca/journals/JIS/VOL13/Styer/styer5.html |via=[[University of Waterloo]]}} Cited in {{harvtxt|Sloane "A055629"}}.</ref>
61626364656667686970717273747576677787980
The number of 10-happy numbers up to 10<sup>''n''</sup> for 1&nbsp;≤&nbsp;''n''&nbsp;≤&nbsp;20 is<ref>{{Cite OEIS |1=A068571 |2=Number of happy numbers <= 10^n}}</ref>
: 3, 20, 143, 1442, 14377, 143071, 1418854, 14255667, 145674808, 1492609148, 15091199357, 149121303586, 1443278000870, 13770853279685, 130660965862333, 1245219117260664, 12024696404768025, 118226055080025491, 1183229962059381238, 12005034444292997294.
8182838384858687888990919293949596979899100

==Happy primes==
A <math>b</math>-happy prime is a number that is both <math>b</math>-happy and [[prime number|prime]]. Unlike happy numbers, rearranging the digits of a <math>b</math>-happy prime will not necessarily create another happy prime. For instance, while 19 is a 10-happy prime, 91&nbsp;=&nbsp;13&nbsp;×&nbsp;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.

===6-happy primes===
In [[base 6]], the 6-happy primes below 1296 = 6<sup>4</sup> are
:211, 1021, 1335, 2011, 2425, 2555, 3351, 4225, 4441, 5255, 5525

===10-happy primes===
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 {{OEIS|id=A035497}}.

The [[palindromic prime]] {{nowrap|10<sup>150006</sup> + {{val|7426247e75000}} + 1}} is a 10-happy prime with {{val|150,007}} digits because the many 0s do not contribute to the sum of squared digits, and {{nowrap|1<sup>2</sup> + 7<sup>2</sup> + 4<sup>2</sup> + 2<sup>2</sup> + 6<sup>2</sup> + 2<sup>2</sup> + 4<sup>2</sup> + 7<sup>2</sup> + 1<sup>2</sup>}} =&nbsp;176, which is a 10-happy number. Paul Jobling discovered the prime in 2005.<ref>{{cite web |url=http://primes.utm.edu/primes/page.php?id=76550 |title=The Prime Database: 10<sup>150006</sup> + 7426247 · 10<sup>75000</sup> + 1 |author=Chris K. Caldwell |work=utm.edu}}</ref>

{{As of|2010}}, the largest known 10-happy prime is 2<sup>42643801</sup>&nbsp;−&nbsp;1 (a [[Mersenne prime]]).{{Dubious|date=December 2017}} Its decimal expansion has {{val|12,837,064}} digits.<ref>{{cite web |url=http://primes.utm.edu/primes/page.php?id=88847 |title=The Prime Database: 2<sup>42643801</sup> − 1 |author=Chris K. Caldwell |work=utm.edu}}</ref>

===12-happy primes===

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, ...

==Programming example==
The examples below implement the perfect digital invariant function for <math>p = 2</math> and a default base <math>b = 10</math> 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 [[Cycle detection|repeating a number]].

A simple test in [[Python (programming language)|Python]] to check if a number is happy:
<syntaxhighlight lang="python">
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
</syntaxhighlight>

==See also==
* [[Arithmetic dynamics#Other areas in which number theory and dynamics interact|Arithmetic dynamics]]
*[[Fortunate number]]
*[[Harshad number]]
*[[Lucky number]]
*[[Perfect digital invariant]]

==References==
{{reflist}}

==Literature==
* {{Cite book
 | last = Guy
 | first = Richard
 | author-link = Richard K. Guy
 | year = 2004
 | title = Unsolved Problems in Number Theory
 |edition=3rd
 | publisher = [[Springer-Verlag]]
 | isbn= 0-387-20860-7
}}

==External links==
* Schneider, Walter: [https://web.archive.org/web/20060204094653/http://www.wschnei.de/digit-related-numbers/happy-numbers.html Mathews: Happy Numbers.]
* {{MathWorld|urlname=HappyNumber|title=Happy Number}}
* [http://nazgul04.ddns.net/happy/happy.php calculate if a number is happy]
* [http://mathforum.org/library/drmath/view/55856.html Happy Numbers] at The Math Forum.
* [https://web.archive.org/web/20180703133816/http://numberphile.com/videos/melancoil.html 145 and the Melancoil] at Numberphile.
* {{cite web|last=Symonds|first=Ria|title=7 and Happy Numbers|url=http://www.numberphile.com/videos/7happy.html|work=Numberphile|publisher=[[Brady Haran]]|access-date=2 April 2013|archive-url=https://web.archive.org/web/20180115215406/http://www.numberphile.com/videos/7happy.html|archive-date=15 January 2018|url-status=dead}}

{{Classes of natural numbers}}
{{Prime number classes}}

{{DEFAULTSORT:Happy Number}}
[[Category:Arithmetic dynamics]]
[[Category:Base-dependent integer sequences]]