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.

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.

Divisibility-based sets of integers Overview Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic Arithmetic number Factorization forms Prime number Composite number Semiprime number Pronic number Sphenic number Square-free number Powerful number Perfect power Achilles number Smooth number Regular number Rough number Unusual number Constrained divisor sums Perfect number Almost perfect number Quasiperfect number Multiply perfect number Hemiperfect number Hyperperfect number Superperfect number Unitary perfect number Semiperfect number Practical number Erdős–Nicolas number With many divisors Abundant number Primitive abundant number Highly abundant number Superabundant number Colossally abundant number Highly composite number Superior highly composite number Weird number Aliquot sequence-related Untouchable number Amicable number Sociable number Betrothed number Other sets Deficient number Friendly number Solitary number Sublime number Harmonic divisor number Frugal number Equidigital number Extravagant number