Tetration

The tetration of a number a by a positive integer b, denoted by $a\uparrow \uparrow b$ or ^ba, is recursively defined by:

$a\uparrow \uparrow 1=a$
$a\uparrow \uparrow (k+1)=a^{a\uparrow \uparrow k}$

Thus we have e.g. $2\uparrow \uparrow 2=4$ and $3\uparrow \uparrow 3=7625597484987$.

Let $P_{n,i}$ be the i-th prime number among all prime numbers $>n$ such that $P_{10,1}=11$, $P_{30,2}=37$ and $P_{100,3}=107$.

Let $MOD(n,m)=n$ modulo $m$.

Further define ${\displaystyle f(k)=\sum _{i=1}^{k}MOD(i\uparrow \uparrow i,P_{k,i})}$.

For example $f(3)=MOD(1\uparrow \uparrow 1,5)+MOD(2\uparrow \uparrow 2,7)+MOD(3\uparrow \uparrow 3,11)=14$.

You are given $f(100)=20245$.

Find $f({10}^6)$.