Rough number

From HandWiki

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:

  • 2-rough numbers: A000027
  • 3-rough numbers: A005408
  • 5-rough numbers: A007310
  • 7-rough numbers: A007775
  • 11-rough numbers: A008364
  • 13-rough numbers: A008365
  • 17-rough numbers: A008366
  • 19-rough numbers: A166061
  • 23-rough numbers: A166063