The greatest prime factor 
of an integer 
is the largest prime factor
of 
. Thus 
in the prime factorization
 |
where 
and the 
are positive integers. For 
, 3, ..., the first few values are 2, 3, 2, 5, 3,
7, 2, 3, 5, 11, 3, 13, 7, 5, ... (OEIS A006530).
The greatest repeated prime factors, meaning the
largest primes whose squares divide 
, for squareful integers
are 2, 2, 3, 2, 2, 3, 2, 2, 5, 3, 2, 2, 3, ... (OEIS A046028).
The integers 
for which 
are precisely those that are not 
-smooth. They begin
2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 20, 21, ... (OEIS A064052).
Their natural density is 
(Schroeppel 1972).
The "unusual numbers" of Greene and Knuth (1990) use the weak condition 
and begin 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, ... (OEIS A063538).
The two conditions differ exactly at the squares of prime
numbers. Finch (2001) also used "
-rough" for unusual numbers. This use differs from
the definition of a rough number, which constrains
every prime factor rather than only the greatest
one.
The complementary 
-smooth numbers, also called "round
numbers," begin 1, 4, 8, 9, 12, 16, 18, 24, 25, 27, ... (OEIS A048098).
and 
."
Aequationes Math. 17, 311-321, 1978.Finch, S. "RE:
Unusual Numbers." 27 Aug 2001. 
." §B46 in 
." Nieuw Arch. Wisk. 1, 113-122,
1953.Schroeppel, R. Item 29 in Beeler, M.; Gosper, R. W.; and Schroeppel,
R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239,
p. 13, Feb. 1972.