Happy number

In Mathematical, a happy number is a Integer which, when one adds the square each one of his Chiffre S, then the squares of the figures of this result and so on until obtaining a number with only one figure, gives 1 for résultat.
On the other hand, the numbers which are not happy are called unhappy numbers.

In a more formal way, one considers a positive entirety $t$, then one defines the continuation $t\_0,\; t\_1,\; t\_2\; \backslash \; ldots\; \backslash ,\; \backslash !$ where $t\_0\; =\; T\; \backslash ,\; \backslash !$ and $t\_\; \{i+1\}$ are equal to the sum of the squares of the figures of $t\_i\; \backslash ,\; \backslash !$. $t$ is known as happy if the continuation leads to 1 starting from a certain number of terms, i.e. for a certain index $i$, $t\_i\; =\; 1\; \backslash ,\; \backslash !$ (Of course, starting from this index, all the $t\_j\; \backslash ,\; \backslash !$ is equal to 1 and the continuation is constant). Thus, 7 is happy, since the associated continuation is:

$t\_1\; =\; 7^2\; =\; 49\; \backslash ,$

$t\_2\; =\; 4^2\; +\; 9^2\; =\; 97\; \backslash ,$

$t\_3\; =\; 9^2\; +\; 7^2\; =\; 130\; \backslash ,$

$t\_4\; =\; 1^2\; +\; 3^2\; +\; 0^2\; =\; 10\; \backslash ,$

$t\_5\; =\; 1^2\; +\; 0^2\; =\; 1\; \backslash ,$

The answer $t$ of the square of a happy number will always give another happy number. As in the preceding continuation.

$t\_1\; =\; 7^2\; =\; 49\; \backslash ,$; 49 = Happy Number

$t\_2\; =\; 4^2\; +\; 9^2\; =\; 97\; \backslash ,$; 97 = Happy Number

$t\_3\; =\; 9^2\; +\; 7^2\; =\; 130\; \backslash ,$; 130 = Happy Number

$t\_4\; =\; 1^2\; +\; 3^2\; +\; 0^2\; =\; 10\; \backslash ,$; 10 = Happy Number

Can imports the starting value in the continuation, if this one is the answer of more a happy small number

A $t$ number is happy if and only if all the members of the continuation defined above are happy. A contrario , a $t$ number is unhappy if and only if all the members of this continuation are unhappy.

The first twenty happy numbers are 1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97 and 100.

See too

• Residue of a natural entirety

External bonds

• happy Numbers on Maths World
• happy Numbers on Mathews

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