Squarefree kernel

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The squarefree kernel of a positive integer $n$ (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)

sqf (n) \equiv rad (n) : = \prod_{\binom{p ∣ n}{p prime}} p .

**A007947: Product of distinct prime factors of $n, 1 \leq n \leq 257 .$**

257 $rad (n)$
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0

In the above graph, notice how the squarefree numbers (with asymptotic density $\frac{1}{ζ (2)} = \frac{6}{π^{2}} \approx 0.60792710185403 \dots$ ) stand out.

A007947 The largest squarefree number dividing n (the squarefree kernel of $n$ , the radical of $n$ )

{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.

r a d (m, n) = r a d (m) \cdot r a d (n), (m, n) = 1 .

Trivial formulas (most trivial first)

$r a d (r a d (n)) = r a d (n)$ (Idempotence^[1])

$q (r a d (n)) = 1$ (where $q (n)$ is the characteristic function of squarefree numbers, or quadratfrei function)

$q (n) = [r a d (n) = n]$ (where $[\cdot]$ is the Iverson bracket)

$Ω (r a d (n)) = ω (r a d (n)) = ω (n)$

$μ (r a d (n)) = {(- 1)}^{ω (r a d (n))} = {(- 1)}^{ω (n)}$

$μ (n) = μ (r a d (n)) \cdot q (n)$

$σ_{0} (r a d (n)) = τ (r a d (n)) = 2^{ω (n)}$

$σ_{1} (r a d (n)) = ψ (r a d (n))$ (Cf. A048250)

where  $ψ (n) \equiv n \cdot \prod_{\binom{p | n}{p p r i m e}} (1 + \frac{1}{p}) = n \cdot \prod_{\binom{p | n}{p p r i m e}} \frac{σ_{1} (p)}{p} = \frac{n}{r a d (n)} \cdot \prod_{\binom{p | n}{p p r i m e}} σ_{1} (p)$  is Dedekind psi function (Cf. A001615)

and thus  $ψ (r a d (n)) = \frac{r a d (n)}{r a d (r a d (n))} \cdot \prod_{\binom{p | r a d (n)}{p p r i m e}} σ_{1} (p) = \prod_{\binom{p | n}{p p r i m e}} σ_{1} (p) = \prod_{\binom{p | r a d (n)}{p p r i m e}} σ_{1} (p) = σ_{1} (r a d (n))$  

and  $ψ (n) = \frac{n}{r a d (n)} \cdot ψ (r a d (n)) = \frac{n}{r a d (n)} \cdot σ_{1} (r a d (n))$

$ψ (r a d (n)) = \frac{ψ (n) \cdot r a d (n)}{n}$ (Cf. A048250)

$ψ (n) \cdot r a d (n) = n \cdot ψ (r a d (n))$ , or $\frac{ψ (n)}{n} = \frac{ψ (r a d (n))}{r a d (n)}$

$ϕ (n) \cdot r a d (n) = n \cdot ϕ (r a d (n))$ , or $\frac{ϕ (n)}{n} = \frac{ϕ (r a d (n))}{r a d (n)}$

where  $ϕ (r a d (n)) = ϕ (\prod_{\binom{p | n}{p p r i m e}} p) = \prod_{\binom{p | n}{p p r i m e}} ϕ (p) = \prod_{\binom{p | n}{p p r i m e}} (p - 1) = r a d (n) \cdot \prod_{\binom{p | n}{p p r i m e}} (1 - \frac{1}{p}) = r a d (n) \cdot \frac{ϕ (n)}{n}$

$n # = r a d (n!)$ (primorial of $n$ )

$\frac{n!}{n #} = \frac{n!}{r a d (n!)}$ (compositorial of $n$ )

n divided by the largest squarefree divisor of n

The n divided by the largest squarefree divisor of n function is

\frac{n}{r a d (n)} = n \cdot \prod_{\binom{p ∣ n}{p p r i m e}} \frac{1}{p} .

This is a multiplicative arithmetic function, e.g.

\frac{m n}{r a d (m n)} = \frac{m}{r a d (m)} \cdot \frac{n}{r a d (n)}, (m, n) = 1 .

Sequences

A003557 $n$ divided by largest squarefree divisor of $n, n \geq 1 .$

{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, ...}

Notes

↑ Idempotence—Wikipedia.org.

[1] Idempotence—Wikipedia.org.

[1]


257 $rad (n)$
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border-right-width: 0; border-top-color: black; border-bottom-color: black; border-left-color: 0; border-right-color: 0; background-color: rgb(201, 218, 237); height: \| 400*254/257 \| px; bottom: 0; width: 3px; margin: 0 0 0 0; z-index: 0;">	<div style="position: absolute; border-style: solid; border-top-width: 1px; border-bottom-width: 1px; border-left-width: 0; border-right-width: 0; border-top-color: black; border-bottom-color: black; border-left-color: 0; border-right-color: 0; background-color: rgb(206, 223, 242); height: \| 400*255/257 \| px; bottom: 0; width: 3px; margin: 0 0 0 0; z-index: 0;">	<div style="position: absolute; border-style: solid; border-top-width: 1px; border-bottom-width: 1px; border-left-width: 0; border-right-width: 0; border-top-color: black; border-bottom-color: black; border-left-color: 0; border-right-color: 0; background-color: rgb(201, 218, 237); height: \| 400*2/257 \| px; bottom: 0; width: 3px; margin: 0 0 0 0; z-index: 0;">	<div style="position: absolute; border-style: solid; border-top-width: 1px; border-bottom-width: 1px; border-left-width: 0; border-right-width: 0; border-top-color: black; border-bottom-color: black; border-left-color: 0; border-right-color: 0; background-color: rgb(206, 223, 242); height: \| 400*257/257 \| px; bottom: 0; width: 3px; margin: 0 0 0 0; z-index: 0;">
0

Squarefree kernel

Contents

Properties

Trivial formulas (most trivial first)

n divided by the largest squarefree divisor of n

Sequences

See also

Notes

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