happy numbers?

Question

WHAT IS A HAPPY NUMBER AND WHY ARE THEY CALLED THAT?WHAT MAKES IT A HAPPY NUMBER AND WHAT IS A UNHAPPY NUMBER AND WHAT MAKES IT A UNHAPPY NUMBER

Answer 1

Here's a link that answers your question. (I hope that makes you happy, lol : )

http://en.wikipedia.org/wiki/Happy_number

Answer 2

A happy number is a mathematical term for a number that eventually reduces to 1 when the following process is used: take the sum of the squares of its digits, and continue iterating this process until it yields 1, or produces an infinite loop. Numbers that are not happy are called unhappy numbers. A computer search up to 1010 suggests that about 15 percent of numbers are happy, though no proof is known.

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

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.

The study of happy numbers is an example of recreational mathematics in that it can involve extensive mathematical knowledge, but the topic is not a central part of serious research.

More formally, given a number n = n0, define a sequence n1, n2, ... where ni + 1 is the sum of the squares of the digits of ni. Then n is happy if and only if this sequence goes to 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. (sequence A007770 in OEIS)


f 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,

n\geq10^{m-1}>81m

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. A computer program can easily verify that in the range 1 to 99, every number is either happy, or goes to the above cycle.

Sets of happy and unhappy numbers exist in every base. These all have similar behavior, such as happy numbers eventually iterating to 1, and all unhappy number sequences leading to infinite loops consisting of values less than 1000b, where b is an accepted sub script for a base.

Similar to the above sequence behavior, any number less than 1000b has the value B3 − 1 and can't have a digit sum any larger than 3 * (B − 1)2, or 3B2 − 6B + 3. Since this is true for all B > 0 any acceptable base will also have repeating unhappy number sequences less than 1000b.

Binary, also known as Base 2, only has happy numbers. All binary numbers larger than 10002 decay into a value equal to or less than 10002, and all such values are happy. Therefore, all numbers in base two are considered happy. This makes base 2 a Happy Base.

Each of the following are in base 2:

{111} -> 11
{11, 101, 110} -> 10
{1, 10, 100, 1000} -> 1

The only known Happy bases are 2 and 4, yet more could exist. Although, it is impossible to have an Unhappy base unless the base itself was infinite, many bases don't even have happy numbers between one and their value 10b. Here are some of the bases without this common property:

(9, 10, 16, 18, 19, 20, 29, 30, 34, ...)

Answer 3

A number is happy if when you follow these steps, you will eventually result to 1.
Steps:
a) Take the squares of the digits of a certain number.
E.g.
in 49, take
4² = 16 and
9² = 81
b) Add the squares of the digits
E.g.
in 49, add
16 + 81 = 97
c) Get again the squares of the digits and add them. Do it all over again until you reach 1 or you will not reach 1.
E.g.
9² + 7² = 81 + 49 = 130
1² + 3² + 0² = 1 + 9 + 0 = 10
1² + 0² = 1 + 0 = 1

Therefore, 49 is a happy number.
^_^

Answer 4

4 8 15 16 23 42

Answer 5

They pretend that 13 is a lucky number .I don't beleive this !

Answer 6

OMG...