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 p 1 r 1 ⋅ p 2 r 2 ⋅ p 3 r 3 ⋅ … p k r k {\displaystyle p_{1}^{r_{1}}\cdot p_{2}^{r_{2}}\cdot p_{3}^{r_{3}}\cdot \dots p_{k}^{r_{k}}} has p i r i ≥ n {\displaystyle p_{i}^{r_{i}}\geq n} for every 1 ≤ i ≤ k {\displaystyle 1\leq i\leq k} (while the condition is p i r i ≤ n {\displaystyle p_{i}^{r_{i}}\leq n} for n-powersmooth numbers), 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.

Sequences

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

See also

Notes

  • Weisstein, Eric W. . MathWorld.
  • "Divisibility, Smoothness and Cryptographic Applications", D. Naccache and I. E. Shparlinski, pp. 115–173 in Algebraic Aspects of Digital Communications, eds. Tanush Shaska and Engjell Hasimaj, IOS Press, 2009, ISBN 9781607500193.