A ''k''-rough number, as defined by Finch in 2001 and 2003, is a positive
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
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''.
[p. 130, Naccache and Shparlinski 2009.]
Examples (after Finch)
#Every odd positive integer is 3-rough.
#Every positive integer that is
congruent
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ...
to 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
*
Buchstab function
The Buchstab function (or Buchstab's function) is the unique continuous function \omega: \R_\rightarrow \R_ defined by the delay differential equation
:\omega(u)=\frac 1 u, \qquad\qquad\qquad 1\le u\le 2,
: (u\omega(u))=\omega(u-1), \qquad u\ge 2.
...
, used to count rough numbers
*
Smooth number
In number theory, an ''n''-smooth (or ''n''-friable) number is an integer whose prime factors are all less than or equal to ''n''. For example, a 7-smooth number is a number whose every prime factor is at most 7, so 49 = 72 and 15750 = 2 × 32 � ...
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, .
The
On-Line Encyclopedia of Integer Sequences
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to t ...
(OEIS)
lists ''p''-rough numbers for small ''p'':
* 2-rough numbers:
A000027
A, or a, is the first letter and the first vowel of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''a'' (pronounced ), plural ''aes' ...
* 3-rough numbers:
A005408
A, or a, is the first letter and the first vowel of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''a'' (pronounced ), plural ''aes' ...
* 5-rough numbers:
A007310
A, or a, is the first letter and the first vowel of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''a'' (pronounced ), plural ''aes' ...
* 7-rough numbers:
A007775
A, or a, is the first letter and the first vowel of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''a'' (pronounced ), plural ''aes' ...
* 11-rough numbers:
A008364
A, or a, is the first letter and the first vowel of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''a'' (pronounced ), plural ''aes' ...
* 13-rough numbers:
A008365
A, or a, is the first letter and the first vowel of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''a'' (pronounced ), plural ''aes' ...
* 17-rough numbers:
A008366
* 19-rough numbers:
A166061
* 23-rough numbers:
A166063
Integer sequences
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