Rough number
Appearance
Ak-rough number,as defined by Finch in 2001 and 2003, is a positiveintegerwhoseprime factorsare all greater than or equal tok.k-roughness has alternately been defined as requiring all prime factors to strictly exceedk.[1]
Examples (after Finch)
[edit]- Every odd positive integer is 3-rough.
- Every positive integer that iscongruentto 1 or 5 mod 6 is 5-rough.
- Every positive integer is 2-rough, since all its prime factors, being prime numbers, exceed 1.
See also
[edit]- Buchstab function,used to count rough numbers
- Smooth number
Notes
[edit]- ^p. 130, Naccache and Shparlinski 2009.
References
[edit]- 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,ISBN9781607500193.
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