Problem 29

Distinct Powers

Consider all integer combinations of $a^{b}$ for $2 \leq a \leq 5$ and $2 \leq b \leq 5$ :

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 $2 \leq a \leq 100$ and $2 \leq b \leq 100$ ?

不同的幂

考虑所有满足 $2 \leq a \leq 5$ 和 $2 \leq b \leq 5$ 的幂 $a^{b}$ ：

$\begin{aligned} 2^{2} = 4, 2^{3} = 8, 2^{4} = 16, 2^{5} = 32 \\ 3^{2} = 9, 3^{3} = 27, 3^{4} = 81, 3^{5} = 243 \\ 4^{2} = 16, 4^{3} = 64, 4^{4} = 256, 4^{5} = 1024 \\ 5^{2} = 25, 5^{3} = 125, 5^{4} = 625, 5^{5} = 3125 \end{aligned}$
如果把这些幂从小到大排列并去重，可以得到如下由 $15$ 个不同的项组成的数列：

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

考虑所有满足 $2 \leq a \leq 100$ 和 $2 \leq b \leq 100$ 的幂 $a^{b}$ ，将它们排列并去重所得到的数列有多少个不同的项？