A15866 | Li Hua and Array

Li Hua wants to solve a problem about $\varphi$ — Euler's totient function. Please recall that $\varphi(x)=\sum\limits_{i=1}^x[\gcd(i,x)=1]$ . $^{\dagger,\ddagger}$

He has a sequence $a_1,a_2,\cdots,a_n$ and he wants to perform $m$ operations:

- "1 $l$ $r$ " ( $1\le l\le r\le n$ ) — for each $x\in[l,r]$ , change $a_x$ into $\varphi(a_x)$ .
- "2 $l$ $r$ " ( $1\le l\le r\le n$ ) — find out the minimum changes needed to make sure $a_l=a_{l+1}=\cdots=a_r$ . In each change, he chooses one $x\in[l,r]$ , change $a_x$ into $\varphi(a_x)$ . Each operation of this type is independent, which means the array doesn't actually change.

Suppose you were Li Hua, please solve this problem.

$^\dagger$ $\gcd(x,y)$ denotes the [greatest common divisor (GCD)](https://en.wikipedia.org/wiki/Greatest_common_divisor) of integers $x$ and $y$ .

$^\ddagger$ The notation $[\textrm{cond}]$ equals $1$ if the condition $\textrm{cond}$ is true, and $0$ otherwise.

The first line contains two integers $n$ and $m$ ( $1\le n,m\le 10^{5}$ ) — the number of elements in the array and the number of operations to process, respectively.

The second line contains $n$ integers $a_{1},a_{2},\cdots ,a_{n}$ ( $1\le a_{i}\le 5\cdot 10^{6}$ ) — the elements of the array.

Next $m$ lines, each line contains three integers $t_{i},l_{i},r_{i}$ ( $t_i\in\{1,2\},1\le l_i\le r_i\le n$ ) — the $i$ -th operation.

For each "2 $l$ $r$ ", output the answer in an separate line.