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# Sequence

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Help how to do this

Cai writes down the list of positive integers, excluding squares and cubes and all perfect powers.  His sequence starts
2, 3, 5, 6, 7, 10, 11, ...
What is the 100th term in Cai's list?

Mar 16, 2023

#1
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Start by counting all the perfect powers of non perfect powers of 2, 3, 5, 6, 7, 10 that are below 100. How many perfect powers we have, how many numbers we need to add onto 100. Then we repeat this process with the numbers from 100 to the new number we get until we finally have no numbers left (it wont be so confusing later on)

First we look at multiples of 2. There is 2^0 (we will only be counting 1 once) 2^2, 2^3, all the way to 2^6 ( 64). There is 6 numbers

Then look at multiples of 3. There is  3^2, 3^3, 3^4 There is 3 numbers

we skip 4 because 2 already includes all of these

From here, 5, 6, 7, and 10 all only have 1 multiple.

We find the total number of multiples under 100 by doing 6 + 3 + 4  = 13

Adding 13 to 100 we get 113.

We find that there are no numbers between 100 and 113 that are perfect powers therefore our answer is, 113

Mar 17, 2023