Consider all integer combinations of $a^b$ for $2 \le a \le 5$ and $2 \le b \le 5$:
$$2^2=4, 2^3=8, 2^4=16, 2^5=32$$$$3^2=9, 3^3=27, 3^4=81, 3^5=243$$$$4^2=16, 4^3=64, 4^4=256, 4^5=1024$$$$5^2=25, 5^3=125, 5^4=625, 5^5=3125$$If they are then placed in numerical order, with any repeats removed, we get the following sequence of 15 distinct terms:
4, 8, 9, 16, 25, 27, 32, 64, 81, 125, 243, 256, 625, 1024, 3125
How many distinct terms are in the sequence generated by $a^b$ for $2 \le a \le 100$ and $2\le b\le 100$?
In [1]:
print(len(set(a**b for a in range(2,101) for b in range(2,101))))
Alternative solution using itertools:
In [2]:
from itertools import product
print(len(set(a**b for a, b in product(range(2, 101), repeat=2))))
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