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

1,4

Links

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

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)

Example

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A005117, A007947, A007948.

Cf. A065463, A065465.

Sequence in context: A320437 A059286 A345940 * A173956 A306078 A284273

Adjacent sequences: A076995 A076996 A076997 * A076999 A077000 A077001

Keyword

nonn,easy

Author

Jon Perry, Nov 28 2002

Status

approved