Function Repository Resource:

GenerateSmoothNumbers

Generate smooth numbers up to some maximum

Contributed by: Sander Huisman

ResourceFunction["GenerateSmoothNumbers"][max]

generates all positive integers up to max whose prime factors are equal to or less than 7.

ResourceFunction["GenerateSmoothNumbers"][max,p]

generates all positive integers up to max whose prime factors are equal to or less than p.

Details

ResourceFunction["GenerateSmoothNumbers"][max,p] generates p-smooth numbers.

ResourceFunction["GenerateSmoothNumbers"][max] generates the so-called humble numbers.

Generate the numbers up to 100 that have factors that are 13 or less:

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Generate the numbers up to 100 that have factors that are 7 or less—so-called humble numbers:

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The second argument does not have to be an integer:

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Check that the prime factors are all below 6.5:

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Smooth numbers can be found by sieving:

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The algorithm used by GenerateSmoothNumbers is much faster though:

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${AbsoluteTiming[ResourceFunction["GenerateSmoothNumbers"][10^5, 7];], AbsoluteTiming[ Select[Range[10^5], FactorInteger[#][[-1, 1]] <= 7 &];]}$

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The argument of GenerateSmoothNumbers can be really large—much larger than practical for a method based on sieving:

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