## Distinct powers

### Problem 29

Published on Friday, 25th October 2002, 06:00 pm; Solved by 67600; Difficulty rating: 5%Consider all integer combinations of *a*^{b} for 2 ≤ *a* ≤ 5 and 2 ≤ *b* ≤ 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 ≤ *a* ≤ 100 and 2 ≤ *b* ≤ 100?