Distinct powers revisited

Consider all integer combinations of a^b for $2\le a\le 5$ and $1\le b\le 5$:

2¹=2, 2²=4, 2³=8, 2⁴=16, 2⁵=32
3¹=3, 3²=9, 3³=27, 3⁴=81, 3⁵=243
4¹=4, 4²=16, 4³=64, 4⁴=256, 4⁵=1024
5¹=5, 5²=25, 5³=125, 5⁴=625, 5⁵=3125

If they are then placed in numerical order, with any repeats removed, we get the following sequence of 18 distinct terms:

2, 3, 4, 5, 8, 9, 16, 25, 27, 32, 64, 81, 125, 243, 256, 625, 1024, 3125

Given two integers $N>1$ and $M>0$, let $R(N,M)$ be the number of distinct terms in the sequence generated by a^b for $2\le a\le N$ and $1\le b\le M$

For example, $R(5,5)=18$, $R({10}^2,{10}^3)=92715$ and $R({10}^5,{10}^6)=99813359409$

Find the last 20 digits of $R({10}^{21},{11}^{21})$

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This problem is a harder version of Project Euler Problem#29 (https://projecteuler.net/problem=29)