Greatest prime factor of n

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The greatest prime factor of an integer

n

is the largest prime number that divides

n

. For example, the greatest prime factor of 44100 is 7 (all larger divisors of 44100 are composite). A prime number is trivially its own greatest prime factor (as well as its own least prime factor). By convention, 1 (which used to be considered prime, but is now called a unit) is sometimes given as its own greatest prime factor. Actually, 1 is the empty product (defined as the multiplicative identity, i.e. 1) of primes.

The density of positive integers with greatest prime factor prime (n) is zero, and is equal to the density of prime (n)-smooth numbers minus the density of prime (n  −  1)-smooth numbers, which are both equal to zero.

Sequences

A006530 Largest noncomposite dividing

n, n   ≥   1

. (Largest prime dividing

n, n   ≥   2

.)

{1, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 5, 2, 17, 3, 19, 5, 7, 11, 23, 3, 5, 13, 3, 7, 29, 5, 31, 2, 11, 17, 7, 3, 37, 19, 13, 5, 41, 7, 43, 11, 5, 23, 47, 3, 7, 5, 17, 13, 53, 3, 11, 7, 19, ...}

See also

• p-smooth numbers
• Largest prime dividing n
• A006530 Gpf (n): greatest prime factor of n (greatest prime dividing n), with

  a (1) = 1

  .

• p-rough numbers
• Smallest prime dividing n
• A020639 Lpf (n): least prime factor of n (least prime dividing n), with

  a (1) = 1

  .