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) Every odd positive integer is 3-rough. Every positive integer that is congruent to 1 or 5 mod 6 is 5-rough. 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 Buchstab function, used to count rough numbers Smooth number
Notes
References Weisstein, Eric W. "Rough Number". MathWorld. Finch's definition from Number Theory Archives "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.