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# Sequences

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I need help with sequences

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 6, 2023

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This list is that of the natural counting numbers, which is free of squares, cubes and perfect powers:

(2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15,  17, 18, 19, 20, 21, 22, 23, 24, 26,  28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113)==100 terms.

In this list, the 100th term is 113.

This list of counting numbers, whose divisors or factors, are square-free, cube-free and perfect power-free.

(2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113, 114, 115, 118, 119, 121, 122, 123, 127, 129, 130, 131, 133, 134, 137, 138, 139, 141, 142, 143, 145, 146, 149, 151, 155, 157, 159, 161, 163, 165, 166)==100 terms.

In this list, the 100th term is 166.

Mar 7, 2023