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Squarefree kernel

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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)


A007947: Product of distinct prime factors of
257  


    

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0  

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, ...}

Properties

The squarefree kernel function (radical function) is a multiplicative arithmetic function, e.g.

Trivial formulas (most trivial first)

  •      (Idempotence[1])



  •      (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





Notes