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Squarefree kernel
From OeisWiki
(Redirected from Squarefree kernel function)
The squarefree kernel of a positive integer (or radical of n, rad(n)) is defined as the product of distinct prime factors of n (i.e. the largest squarefree number dividing n)
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In the above graph, notice how the squarefree numbers (with asymptotic density ) stand out.
A007947 The largest squarefree number dividing n (the squarefree kernel of , the radical of )
- {1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 6, 5, 26, 3, 14, 29, 30, 31, 2, 33, 34, 35, 6, 37, 38, 39, 10, 41, 42, 43, 22, 15, 46, 47, 6, 7, 10, 51, 26, 53, 6, 55, 14, 57, 58, ...}
Contents
Properties
The squarefree kernel function (radical function) is a multiplicative arithmetic function, e.g.
Trivial formulas (most trivial first)
- (where is the characteristic function of squarefree numbers, or quadratfrei function)
- (where is the Iverson bracket)
- (Cf. A048250)
where is Dedekind psi function (Cf. A001615) and thus and
- (Cf. A048250)
- , or
- , or
where
- (primorial of )
- (compositorial of )
n divided by the largest squarefree divisor of n
The n divided by the largest squarefree divisor of n function is
This is a multiplicative arithmetic function, e.g.
Sequences
A003557 divided by largest squarefree divisor of
- {1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 2, 1, 1, 1, 8, 1, 3, 1, 2, 1, 1, 1, 4, 5, 1, 9, 2, 1, 1, 1, 16, 1, 1, 1, 6, 1, 1, 1, 4, 1, 1, 1, 2, 3, 1, 1, 8, 7, 5, 1, 2, 1, 9, 1, 4, 1, 1, 1, 2, 1, 1, 3, 32, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 5, 2, ...}
See also
- A036691 Compositorial numbers: product of first composite numbers.
- A049614 Compositorial of ( divided by its squarefree kernel.)
- A007947 Largest squarefree number dividing (the squarefree kernel, or radical, of ).
- {{Squarefree kernel}} ({{radical}}, {{rad}}) arithmetic function template
- Distinct prime factors of n (prime factors of n (without multiplicity))
- Number of distinct prime factors of n (little omega(n))
- Sum of distinct prime factors of n (sodpf(n))
- Product of distinct prime factors of n (radical of n, rad(n)) (squarefree kernel of n)
- Prime factors of n (with multiplicity)
- Number of prime factors of n (with multiplicity) (big Omega(n))
- Sum of prime factors of n (with multiplicity) (sopf(n)) (integer log of n)
- Product of prime factors of n (with multiplicity) (n, positive integers)