Happy_number

Happy number

Numbers with a certain property involving recursive summation

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 $1^{2}+3^{2}=10$ , and $1^{2}+0^{2}=1$ . On the other hand, 4 is not a happy number because the sequence starting with $4^{2}=16$ and $1^{2}+6^{2}=37$ eventually reaches $2^{2}+0^{2}=4$ , 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.

Tree showing all happy numbers up to 100

More generally, a $b$ -happy number is a natural number in a given number base $b$ that eventually reaches 1 when iterated over the perfect digital invariant function for $p=2$ .^[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).

Happy numbers and perfect digital invariants

Formally, let $n$ be a natural number. Given the perfect digital invariant function

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}

.

for base $b>1$ , a number $n$ is $b$ -happy if there exists a $j$ such that $F_{2,b}^{j}(n)=1$ , where $F_{2,b}^{j}$ represents the $j$ -th iteration of $F_{2,b}$ , and $b$ -unhappy otherwise. If a number is a nontrivial perfect digital invariant of $F_{2,b}$ , then it is $b$ -unhappy.

For example, 19 is 10-happy, as

F_{2,10}(19)=1^{2}+9^{2}=82

F_{2,10}^{2}(19)=F_{2,10}(82)=8^{2}+2^{2}=68

F_{2,10}^{3}(19)=F_{2,10}(68)=6^{2}+8^{2}=100

F_{2,10}^{4}(19)=F_{2,10}(100)=1^{2}+0^{2}+0^{2}=1

For example, 347 is 6-happy, as

F_{2,6}(347)=F_{2,6}(1335_{6})=1^{2}+3^{2}+3^{2}+5^{2}=44

F_{2,6}^{2}(347)=F_{2,6}(44)=F_{2,6}(112_{6})=1^{2}+1^{2}+2^{2}=6

F_{2,6}^{3}(347)=F_{2,6}(6)=F_{2,6}(10_{6})=1^{2}+0^{2}=1

There are infinitely many $b$ -happy numbers, as 1 is a $b$ -happy number, and for every $n$ , $b^{n}$ ( $10^{n}$ in base $b$ ) is $b$ -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.

Happy bases

Unsolved problem in mathematics:

Are base 2 and base 4 the only bases that are happy?

(more unsolved problems in mathematics)

A happy base is a number base $b$ where every number is $b$ -happy. The only happy integer bases less than 5×10⁸ are base 2 and base 4.^[3]

Specific b-happy numbers

4-happy numbers

For $b=4$ , the only positive perfect digital invariant for $F_{2,b}$ is the trivial perfect digital invariant 1, and there are no other cycles. Because all numbers are preperiodic points for $F_{2,b}$ , all numbers lead to 1 and are happy. As a result, base 4 is a happy base.

6-happy numbers

For $b=6$ , the only positive perfect digital invariant for $F_{2,b}$ 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 $F_{2,b}$ , 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 numbers

For $b=10$ , the only positive perfect digital invariant for $F_{2,b}$ 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 $F_{2,b}$ , 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 10ⁿ 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.

Happy primes

A $b$ -happy prime is a number that is both $b$ -happy and prime. Unlike happy numbers, rearranging the digits of a $b$ -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.

Programming example

The examples below implement the perfect digital invariant function for $p=2$ and a default base $b=10$ 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

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This article uses material from the Wikipedia article Happy_number, and is written by contributors. Text is available under a CC BY-SA 4.0 International License; additional terms may apply. Images, videos and audio are available under their respective licenses.

[1] [1]
"Sad Number". Wolfram Research, Inc. Retrieved 16 September 2009.

[2] [2]
Gilmer, Justin (2013). "On the Density of Happy Numbers". Integers. 13 (2): 2. arXiv:1110.3836. Bibcode:2011arXiv1110.3836G.

[3] [3]
Sloane, N. J. A. (ed.). "Sequence A161872 (Smallest unhappy number in base n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

[4] [4]
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.

[5] [5]
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.

[6] [6]
Pan, Hao (2006). "Consecutive Happy Numbers". arXiv:math/0607213.

[Sloane-A055629-7] [7]
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.

[8] [8]
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".

[9] [9]
Sloane, N. J. A. (ed.). "Sequence A068571 (Number of happy numbers <= 10^n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

[10] [10]
Chris K. Caldwell. "The Prime Database: 10¹⁵⁰⁰⁰⁶ + 7426247 · 10⁷⁵⁰⁰⁰ + 1". utm.edu.

[11] [11]
Chris K. Caldwell. "The Prime Database: 2^42643801 − 1". utm.edu.

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Happy_number

Happy number

Happy numbers and perfect digital invariants

Natural density of b-happy numbers

Happy bases

Specific b-happy numbers

4-happy numbers

6-happy numbers

10-happy numbers

Happy primes

6-happy primes

10-happy primes

12-happy primes

Programming example

See also

References

Literature

External links

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