Happy number

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Tree showing all happy numbers up to 100, with 130 seen with 13 and 31

In number theory, a happy number is a number which eventually reaches 1 when the number is 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 function for <math>p = 2</math>.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</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" Template:Harvcol.

Happy numbers and perfect digital invariantsEdit

Template:See also Formally, let <math>n</math> be a natural number. Given the 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 iteration of <math>F_{2, b}</math>, and <math>b</math>-unhappy otherwise. If a number is a nontrivial perfect digital invariant of <math>F_{2, b}</math>, then it is <math>b</math>-unhappy.

For example, 19 is 10-happy, as

For example, 347 is 6-happy, as

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 numbersEdit

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>Template:Cite journal</ref>

Happy basesEdit

Template:Unsolved A happy base is a number base <math>b</math> where every number is <math>b</math>-happy. The only happy integer bases less than Template:Val are base 2 and base 4.<ref>Template:Cite OEIS</ref>

Specific b-happy numbersEdit

4-happy numbersEdit

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 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 numbersEdit

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 = 6⁴ 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

10-happy numbersEdit

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 only cycle is the eight-number cycle

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

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.

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.<ref>Template:Cite OEIS</ref> The first set of three consecutive is 1880, 1881, and 1882.<ref>Template:Cite OEIS</ref> It has been proven that there exist sequences of consecutive happy numbers of any natural number length.<ref>Template:Cite arXiv</ref> The beginning of the first run of at least n consecutive 10-happy numbers for n = 1, 2, 3, ... is<ref name="Sloane-A055629">Template:Cite OEIS</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>Template:Cite journal Cited in Template:Harvtxt.</ref>

The number of 10-happy numbers up to 10ⁿ for 1 ≤ n ≤ 20 is<ref>Template:Cite OEIS</ref>

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

Happy primesEdit

A <math>b</math>-happy prime is a number that is both <math>b</math>-happy and 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 = 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.

6-happy primesEdit

In base 6, the 6-happy primes below 1296 = 6⁴ are

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

10-happy primesEdit

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 Template:Nowrap is a 10-happy prime with Template:Val digits because the many 0s do not contribute to the sum of squared digits, and Template:Nowrap = 176, which is a 10-happy number. Paul Jobling discovered the prime in 2005.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Template:As of, the largest known 10-happy prime is 2^42643801 − 1 (a Mersenne prime).{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= Template:Fix }} Its decimal expansion has Template:Val digits.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

12-happy primesEdit

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 exampleEdit

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 repeating a number.

A simple test in 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>

In popular cultureEdit

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.
In the Doctor Who episode 42, a sequence of happy primes is the password to open a door.