Problem 29

Distinct powers

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 $2 \le b \le 100$?

考虑所有满足$2 \le a \le 5$和$2 \le b \le 5$的整数组合所生成的幂$a^b$：

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

如果把这些幂从小到大排列并去重，我们得到以下由$15$个不同的项组成的序列：

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

考虑所有满足$2 \le a \le 100$和$2 \le b \le 100$的整数组合所生成的幂$a^b$，将它们排列并去重所得到的序列有多少个不同的项？