Rough number

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A k-rough number, as defined by Finch in 2001 and 2003, is a positive integer whose prime factors are all greater than or equal to k.


  1. Every odd positive integer (except 1) is 3-rough.
  2. Every positive integer (except 1) that is congruent to 1 or 5 mod 6 is 5-rough.


The number 1 gives rise to the following "trivial" exceptions:

  1. The number 1 is not k-rough, for any positive integer k, since it has no prime factors.
  2. Every positive integer (except 1) is 2-rough, since all its prime factors, being prime numbers, exceed 1.

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