Ak-rough number, as defined by Finch in 2001 and 2003, is a positiveinteger whoseprime factors are all greater than or equal tok.k-roughness has alternately been defined as requiring all prime factors to strictly exceedk.^[1]

1. Every odd positive integer is 3-rough.
2. Every positive integer that iscongruent 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.

Likepowersmooth numbers, we define "n-powerrough numbers" as the numbers whoseprime factorization${\displaystyle p_1^{r_1}\cdot p_2^{r_2}\cdot p_3^{r_3}\cdot \dots p_k^{r_k}}$ has${\displaystyle p_i^{r_i}\ge n}$ for every${\displaystyle 1\le i\le k}$ (while the condition is${\displaystyle p_i^{r_i}\le n}$ forn-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.

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

• 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 inAlgebraic Aspects of Digital Communications, eds. Tanush Shaska and Engjell Hasimaj, IOS Press, 2009,ISBN 9781607500193.

TheOn-Line Encyclopedia of Integer Sequences (OEIS)listsp-rough numbers for smallp:

• 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

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