Rough number

Short description: Positive integer with large prime factors

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 $p_1^{r_1}\cdot p_2^{r_2}\cdot p_3^{r_3}\cdot \dots p_k^{r_k}$ has $p_i^{r_i}\ge n$ for every $1\le i\le k$ (while the condition is $p_i^{r_i}\le n$ 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

References

• Weisstein, Eric W.. "Rough Number". http://mathworld.wolfram.com/RoughNumber.html. 
• 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.

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