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.^[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.

Powerrough numbers

Like powersmooth numbers, we define "n-powerrough numbers" as the numbers whose prime factorization ${\displaystyle p_{1}^{r_{1}}\cdot p_{2}^{r_{2}}\cdot p_{3}^{r_{3}}\cdot \dots p_{k}^{r_{k}}}$ has ${\displaystyle p_{i}^{r_{i}}\geq n}$ for every ${\displaystyle 1\leq i\leq k}$ (while the condition is ${\displaystyle p_{i}^{r_{i}}\leq n}$ 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.

See also

• Buchstab function, used to count rough numbers
• Smooth number