Arithmetic Integral

Number Theory Level 5

$\Large \displaystyle\int n\text{ }dx \neq nx?$

In calculus, when you take the derivative of a constant you get zero as an answer. In number theory, there is something called the arithmetic derivative which allows you to differentiate a number and get a nonzero answer. The arithmetic derivative works as follows.

Where $n'$ denotes the arithmetic derivative of $n$ :

$p' = 1$ for all primes $p$

$(ab)'=a'b+ab'$

$0'=1'=0$

For example, $6'=(2\times3)'=(2')(3)+(2)(3')=(1)(3)+(2)(1)=5$

While you clearly get a single answer when taking the arithmetic derivative of a number, multiple values of $n$ can lead to the same $n'$ . Let us define the arithmetic integral, denoted $\int n$ , as the function that when given a value of $n$ returns a set of all the positive integers $m$ such that $m'=n$ . What is the value of $n$ , where $1<n<100$ , that gives us the set with the largest dimensions? In other words, what is the value of $n$ that has the most solutions $m$ to the equation $m'=n$ ?

This is a member of a set of problems on the Arithmetic Derivative .

The answer is 90.

2 solutions

Patrick Corn
Jul 9, 2015

I cheated! https://oeis.org/A099302/list

Kazem Sepehrinia
Jul 10, 2015

If $n$ has a representation in the form $p_2 p_3 ... p_k+ p_1 p_3 ... p_k + p_1 p_2 ... p_{k-1}$ for distinct primes $p_1, p_2, ..., p_k$ and $k \ge 2$ , then $m=p_1 p_2 ... p_k$ is a solution to the equation $m'=n$ . I think these kind of solutions are easy to seek and even easier for $k=2$ where $n$ can be represented in the form $n=p+q$ and $m=pq$ is a solution to $m'=n$ .

For example $m=6$ is a solution to the equation $m'=5$ , because $6'=2' \times 3+ 3'\times 2=3+2=5$ Or $m=30$ is an answer for the equation $m'=31$ because $30'=2 \times 3+2 \times 5+3 \times 5=31$ For $n<100$ the equation $m'=n$ has maximum possible values of answers when $n=90$ , because $90$ can be represented as sum of two primes in $9$ ways: $\displaystyle \begin{array}{c}\\ 90 &&=7+83=11+79=17+73=19+71 \\ &&=23+67=29+61=31+59=37+53 \\ &&=43+47 \end{array}$ I think it will be too difficult to find all of solutions to the equation $m'=n$ .

Arithmetic Integral

This is a member of a set of problems on the Arithmetic Derivative .

The answer is 90.

2 solutions

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