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 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

• 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, .