arithmetic derivative

The arithmetic derivative $n^{\prime }$ of a natural number $n$ is defined by the following rules:

• $p^{\prime }=1$ for any prime $p$.

• ${(ab)}^{\prime }=a^{\prime }b+a{b}^{\prime }$ for any $a,b\in \mathbb{N}$ (Leibniz rule).

To define the arithmetic derivative of a negative number, we first note that $1^{\prime }=0$ by the Leibniz rule ($1^{\prime }={(1\cdot 1)}^{\prime }=1\cdot 1^{\prime }+1^{\prime }\cdot 1=2\cdot 1^{\prime }$, so $1^{\prime }=0$), and further that we must have

$0=1^{\prime }={\left((-1)\cdot (-1)\right)}^{\prime }=-2\cdot {(-1)}^{\prime },$

so ${(-1)}^{\prime }=0$. The product rule now requires that we define ${(-n)}^{\prime }=-(n^{\prime })+n{(-1)}^{\prime }=-(n^{\prime })$.

Further, we can extend this definition to rational numbers by insisting that the quotient rule holds, i.e. for a prime $p$ we should have

$0=1^{\prime }={\left(p\cdot {\displaystyle \frac{1}{p}}\right)}^{\prime }={\left({\displaystyle \frac{1}{p}}\right)}^{\prime }p+{\displaystyle \frac{1}{p}},$

giving us that

$${\left(\frac{1}{p}\right)}^{\prime }=-\frac{1}{p^2},$$

i.e. the usual quotient rule from calculus. We now complete the definition by extending multiplicatively (i.e. using the Leibniz rule).

The arithmetic derivatives for the first few positive integers are 0, 1, 1, 4, 1, 5, 1, 12, 6, 7, 1, 16, 1, 9, 8, 32, 1, 21, 1, 24, 10, 13, 1, 44, 10, etc.

As a consequence of $p^{\prime }=1$ for a prime $p$, the arithmetic derivative of a semiprime (whether squarefree or not) works out to ${(pq)}^{\prime }=p^{\prime }q+p{q}^{\prime }=1p+q1=p+q$. For example, the arithmetic derivative of 10 is 7, which is 2 plus 5.

The only cases of $n^{\prime }=n$ for are 0, 4, 27.

$n$	$n^{\prime }$	$n$	$n^{\prime }$	$n$	$n^{\prime }$	$n$	$n^{\prime }$	$n$	$n^{\prime }$	$n$	$n^{\prime }$	$n$	$n^{\prime }$	$n$	$n^{\prime }$	$n$	$n^{\prime }$	$n$	$n^{\prime }$
0	0	10	7	20	24	30	31	40	68	50	45	60	92	70	59	80	176	90	123
1	0	11	1	21	10	31	1	41	1	51	20	61	1	71	1	81	108	91	20
2	1	12	16	22	13	32	80	42	41	52	56	62	33	72	156	82	43	92	96
3	1	13	1	23	1	33	14	43	1	53	1	63	51	73	1	83	1	93	34
4	4	14	9	24	44	34	19	44	48	54	81	64	192	74	39	84	124	94	49
5	1	15	8	25	10	35	12	45	39	55	16	65	18	75	55	85	22	95	24
6	5	16	32	26	15	36	60	46	25	56	92	66	61	76	80	86	45	96	272
7	1	17	1	27	27	37	1	47	1	57	22	67	1	77	18	87	32	97	1
8	12	18	21	28	32	38	21	48	112	58	31	68	72	78	71	88	140	98	77
9	6	19	1	29	1	39	16	49	14	59	1	69	26	79	1	89	1	99	75

Title	arithmetic derivative
Canonical name	ArithmeticDerivative
Date of creation	2013-03-22 13:35:09
Last modified on	2013-03-22 13:35:09
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	14
Author	PrimeFan (13766)
Entry type	Definition
Classification	msc 11Z05
Related topic	Prime