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.[1]

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.

See also


  1. ^ p. 130, Naccache and Shparlinski 2009.


The On-Line Encyclopedia of Integer Sequences (OEIS) lists p-rough numbers for small p: