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# example of arithmetic derivative

Consider the natural number $6$. Using the rules of the arithmetic derivative we get:

$6^{{\prime}}=(2\cdot 3)^{{\prime}}=2^{{\prime}}\cdot 3+2\cdot 3^{{\prime}}=1%
\cdot 3+2\cdot 1=5$

Below is a list of the 10 first natural numbers and their first and second arithmetic derivatives:

$n$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|

$n^{{\prime}}$ | $0$ | $1$ | $1$ | $4$ | $1$ | $5$ | $1$ | $12$ | $6$ | $7$ |

$n^{{\prime\prime}}$ | $0$ | $0$ | $0$ | $4$ | $0$ | $1$ | $0$ | $16$ | $5$ | $1$ |

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