The increasing sequence 2, 3, 5, 6, 7, 10. consists of positive integers that are neither the square nor the cube of a positive integer. Let the 1000th term of this sequence be N. Find the sum of the digits of N.
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Answer: Using the inclusion-exclusion principle, the number of positive integers from 1 to N that are in your sequence is
f(N)=N−⌊N1/2⌋−⌊N1/3⌋−⌊N1/5⌋+⌊N1/6⌋+⌊N1/10⌋+⌊N1/15⌋−⌊N1/30⌋
Since most numbers are not powers, we might start by computing f(1000)=960. We are short by 40, so next try f(1040)=999. Pretty close! The next value is f(1041)=1000. So your answer is 1041.
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