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Problem 29

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Distinct Powers

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

$$\begin{array}{rrrr}
2^2=4, &2^3=8, &2^4=16, &2^5=32\\
3^2=9, &3^3=27, &3^4=81, &3^5=243\\
4^2=16, &4^3=64, &4^4=256, &4^5=1024\\
5^2=25, &5^3=125, &5^4=625, &5^5=3125
\end{array}$$

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 $2 \le b \le 100$?

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不同的幂

考虑所有满足$2 \le a \le 5$和$2 \le b \le 5$的幂$a^b$：

$$\begin{array}{rrrr}
2^2=4, &2^3=8, &2^4=16, &2^5=32\\
3^2=9, &3^3=27, &3^4=81, &3^5=243\\
4^2=16, &4^3=64, &4^4=256, &4^5=1024\\
5^2=25, &5^3=125, &5^4=625, &5^5=3125
\end{array}$$

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$2 \le a \le 100$和$2 \le b \le 100$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？

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