A076998 Difference between cubefree and squarefree components of n.

0, 0, 0, 2, 0, 0, 0, 2, 6, 0, 0, 6, 0, 0, 0, 2, 0, 12, 0, 10, 0, 0, 0, 6, 20, 0, 6, 14, 0, 0, 0, 2, 0, 0, 0, 30, 0, 0, 0, 10, 0, 0, 0, 22, 30, 0, 0, 6, 42, 40, 0, 26, 0, 12, 0, 14, 0, 0, 0, 30, 0, 0, 42, 2, 0, 0, 0, 34, 0, 0, 0, 30, 0, 0, 60, 38, 0, 0, 0, 10, 6, 0, 0, 42, 0, 0, 0, 22, 0, 60, 0, 46

Offset: 1

Author

Jon Perry, Nov 28 2002

Keywords

• nonn
• easy

Examples

			a(4)=2 as cubefree(4)=4 and squarefree(4)=2. 4-2=2
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A005117, A007947, A007948.

Cf. A065463, A065465.

Programs

Mathematica

a[n_] := Module[{f = FactorInteger[n]}, Times @@ (First[#]^Min[Last[#], 2] & /@ f) - Times @@ (First[#] & /@ f)]; Array[a, 100] (* Amiram Eldar, Sep 24 2023 *)

PARI

rad(n)=local(p,i); p=factor(n)[,1]; prod(i=1,length(p),p[i])
rad2(n)=local(p,pn,i); p=factor(n)[,1]; pn=factor(n)[,2]; prod(i=1,length(p),p[i]^min(2,pn[i]))
for (k=1,100,print1(rad2(k)-rad(k)", "))

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^min(f[i,2],2)) - vecprod(f[,1]);} \\ Amiram Eldar, Sep 24 2023

Formula

a(n) = A007948(n) - A007947(n). - Antti Karttunen, Jul 21 2018

From Amiram Eldar, Sep 24 2023: (Start)

a(n) >= 0, with equality if and only if n is squarefree (A005117).

Sum_{k=1..n} a(k) ~ (1/2) * c * n^2, where c = A065465 - A065463 = 0.17707163872600518419... . (End)