Rough number


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. k-roughness has alternately been defined as requiring all prime factors to strictly exceed k.

Examples (after Finch)

  1. Every odd positive integer is 3-rough.
  2. Every positive integer that is congruent to 1 or 5 mod 6 is 5-rough.
  3. Every positive integer is 2-rough, since all its prime factors, being prime numbers, exceed 1.

    Powerrough numbers

Like powersmooth numbers, we define "n-powerrough numbers" as the numbers whose prime factorization has for every , e.g. every positive integer is 2-powerrough, 3-powerrough numbers are exactly the numbers not 2 mod 4, 4-powerrough numbers are exactly the numbers neither 2 mod 4 nor 3, 6 mod 9, 5-powerrough numbers are exactly the numbers neither 2, 4, 6 mod 8 nor 3, 6 mod 9, etc.