CF1797E.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) 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.

Denote $\varphi^k(x)=\begin{cases}x,&k=0\\\varphi(\varphi^{k-1}(x)),&k > 0\end{cases}$ .

At first, $a=[8,1,6,3,7]$ .

To make sure $a_1=a_2=a_3=a_4=a_5$ , we can change $a$ to $a'=[\varphi^3(8),\varphi^0(1),\varphi^2(6),\varphi^2(3),\varphi^3(7)]=[1,1,1,1,1]$ , using $3+0+2+2+3=10$ changes.

To make sure $a_3=a_4$ , we can change $a$ to $a'=[\varphi^0(8),\varphi^0(1),\varphi^1(6),\varphi^1(3),\varphi^0(7)]=[8,1,2,2,7]$ , using $0+0+1+1+0=2$ changes.

After "1 $1$ $3$ ", $a$ is changed to $a=[\varphi^1(8),\varphi^1(1),\varphi^1(6),\varphi^0(3),\varphi^0(7)]=[4,1,2,3,7]$ .

To make sure $a_3=a_4$ , we can change $a$ to $a'=[\varphi^0(4),\varphi^0(1),\varphi^0(2),\varphi^1(3),\varphi^0(7)]=[4,1,2,2,7]$ , using $0+0+0+1+0=1$ change.