Arithmetic derivative

Given a natural number $n$ , its arithmetic derivative $d(n)$ is defined as follows:

$d(0) = d(1) = 0$ .
If $n$ is prime, then $d(n) = 1$ .
Let $n = x \cdot y$ , with $1 < x, y < n$ . Then $d(n) = x \cdot d(y) + y \cdot d(x)$ .

For instance, $d(4) = 2d(2) + 2d(2) = 2 + 2 = 4$ , and $d(6) = 3d(2) + 2d(3) = 3 + 2 = 5$ . It can be proven that this definition is consistent. For example, $d(12) = 4d(3) + 3d(4) = 4 + 12 = 16$ , and also $d(12) = 6d(2) + 2d(6) = 6 + 10 = 16$ .

We say that $f$ is a fixed point of $d(n)$ if $d(f) = f$ . For instance, 0 and 4 are fixed points. Given $\ell$ and $r$ , can you compute the number of fixed points of $d(n)$ in $[\ell..r]$ ?

Input

Input consists of several cases, each one with $\ell$ and $r$ , with $0 \le \ell \le r \le 10^{18}$ .

Output

For each case, print the number of fixed points of $d(n)$ in $[\ell..r]$ .

Problem information

Author: Salvador Roura

Generation: 2026-01-25T12:08:07.407Z