project euler problem 29数的多次幂的重复个数

Distinct powers

Problem 29

Consider all integer combinations of ab for 2

a

5
and 2

b

5:

22=4, 23=8, 24=16, 25=32

32=9, 33=27, 34=81, 35=243

42=16, 43=64, 44=256, 45=1024

52=25, 53=125, 54=625, 55=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 ab for 2

a

100
and 2

b

100?

Answer: 9183 Completed on Sun, 20 Oct 2013, 06:26

刚开始想到的肯定是从1到10其平方之后有相同的就减掉啦！

2 4 8 16 32 64 //这些数都是从2到10平方到4次方之后可能会重复的数，当他们各自以相同的n次方后就会相同的，

3 9 27 81 //比如25^2到50次方都与(2^2)^2到50次方都相同，故相同的有49个

5 25

6 36

7 49

10 100

所以这些数相同的有：

4 --> 49

8 --> 49

9 --> 49

16 --> 58

25 --> 49

27 --> 49

32 --> 48

36 --> 49

49 --> 49

64 --> 62

81 --> 58

100 --> 49 总共有 618 个重复的。



所以答案为：99*99-618=9183.