happy number (happy number) has the following characteristics: In the given carry system, the number has all the digits square sum, receive the new number

and find the square sum of all the digits again. The final result must be 1.

Example, take decimal as an example:

2 8 → 2^2+8^2=68 → 6^2+8^2=100 → 1^2+0^2+0^2=1

3 2 → 3^2+2^2=13 → 1^2+3^2=10 → 1^2+0^2=1

3 7 → 3^2+7^2=58 → 5^2+8^2=89 → 8^2+9^2=145 → 1^2+4^2+5^2=42 → 4^2+2^2=20 → 2^2+0^2=4 → 4^2=16 → 1^2+6^2=37……

Therefore 28 and 32 are happy numbers, and 37 repeats during 37's Calculator, continuing the Calculator Results will only be the above number loop, and no 1 will occur, so 37 is not a happy Number.

is not a happy number number is called an unhappy number (unhappy number), the number number of all unhappy bits square sum Calculator, Finally, it will enter the 4 → 16 → 37 → 58 → 89 → 145 → 42 → 20 → 4 cycle.

In the decimal, the happiness number up to 100 is (number column A00770 in OEIS) : 1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97, 100.

Maybe We can find something more interesting in less than 10's. So there's no letter in the Number. 167 is 5 times the number of 9's, then in a divisible 9's base, the last digit of the Number is 5, which looks much more festive than the clumsy 7. (Of course, that's only if We're used to our decimal eyes, and 5's doesn't mean what We think it means in base 9.) In base 9, 167 is written as 205, but I personally prefer 25 in base 81, which is very concise.

Studying 167 under different bases leads to another interesting fact: 167 is one strictly non-palindromic number; in other words, it cannot be written as a palindromic number in any one base between 2 and 165 (the Number is exactly the same when read forwards and backwards). (The reason We stop in base 165 is that it is 167-2, and any one Number n in base n-1 both are palindromic numbers, so it looks like both are 11's formality.) So far, We do not know the strict non-palindromic number number order, but one non-palindromic number under 167's is 179, and the one after that is 223.

These characteristics listed above are completely enough to prove the necessity of holding one celebration, in addition, 167 is also one safety element number, one very cototient number, one total cycle number. I particularly like the last one: this means that there is one 166-bit Number, and it cycles through each multiple number both are Number. in other words, when you multiply this number by one whole number, the receive product is exactly the original number Number, in the same order, but with a different starting point, Example 142857×2=285714.