Project Euler 29

题目

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 $2 \le b \le 100$?

解决方案

使用Python计算大数的特点，可以直接将所有数计算出来。

另外一种方案是，把每个数对$e$取对数，然后把$b\ln a$计算出来。不过这种做法的代价是需要特别注意浮点数的误差。

代码

N = 100
ans = len(set(a ** b for a in range(2, N + 1) for b in range(2, N + 1)))
print(ans)