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Pattern?

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I was digging through some very old google docs and deleting them to make space to add a 1.5M char doc to the stock when I came across one titled 'Prime Pattern?'. I couldn't resist opening it and I immediately remembered what it held.

When I created it a year ago, I asked myself 'what is the smallest number that has the first \(n\) integers as some of its factors?' I came up with a list of smallest numbers with the first \(n\) integers as its factors up to \(n=150\). The first 25 are:

1, 2, 6, 12, 60, 60, 420, 840, 2520, 2520, 2520, 27720, 360360, 360360, 360360, 720720, 12252240, 12252240, 232792560, 232792560, 232792560, 232792560, 5354228880, 5354228880, 26771144400.

However, listing the smallest positive integer that isn't a factor to each of the above will get me the following:

2, 3, 4, 5, 7, 7, 8, 9, 11, 11, 13, 13, 16, 16, 16, 17, 19, 19, 23, 23, 23, 23, 25, 25, 27.

I noticed that every single of these numbers were in the form of \(p^k\), where \(p\) is prime and \(k\) is any natural number. Is there a way to exploit this into generating a list of numbers in the form of \(p^k\)?

Jan 23, 2019