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The sequence 2, 3, 5, 6, 7, 10, 11,  contains all the positive integers from least to greatest that are neither squares nor cubes. What is the 500th  term of the sequence?

 Dec 23, 2021
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We can start witht the number 500 and count all of the squares/cubes under that.

222 (484) is the closest we can get to 500 without going over it.
73 (343) os the closest we can get to 500 without going over it.

22+7 is 29, which is how many we got rid of (almost, but not really)

Since 1 is both a square and a cube, we need to subtract 1 for the double count. 
Since 64 is both a square and a cube, we need to subtract another 1 for the double count.

So to compensate for each square and cube not included, we can add 27 (29-2) to 500
The 500th term would be 527, but 512 is 8^3, so we also have to compensate for that.

527+1=528

 Dec 23, 2021

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