A056552 Powerfree kernel of cubefree part of n.

1, 2, 3, 2, 5, 6, 7, 1, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 3, 5, 26, 1, 14, 29, 30, 31, 2, 33, 34, 35, 6, 37, 38, 39, 5, 41, 42, 43, 22, 15, 46, 47, 6, 7, 10, 51, 26, 53, 2, 55, 7, 57, 58, 59, 30, 61, 62, 21, 1, 65, 66, 67, 34, 69, 70, 71, 3, 73, 74, 15, 38, 77

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Henry Bottomley, Some Smarandache-type multiplicative sequences.

Formula

a(n) = A007947(A050985(n)) = A019555(A050985(n)) = n/(A053150(n)*A000189(n)) = A019555(n)/A053150(n) = A056551(n)^(1/3).

If n = Product_{j} Pj^Ej then a(n) = Product_{j} Pj^Fj, where Fj = 0 if Ej is 0 or a multiple of 3 and Fj = 1 otherwise.

Multiplicative with a(p^e) = p^(if 3|e, then 0, else 1). - Mitch Harris, Apr 19 2005

Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(6)/2) * Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4) = 0.3480772773... . - Amiram Eldar, Oct 28 2022

Dirichlet g.f.: zeta(3*s) * Product_{p prime} (1 + 1/p^(s-1) + 1/p^(2*s-1)). - Amiram Eldar, Sep 16 2023

Example

a(32) = 2 because cubefree part of 32 is 4 and powerfree kernel of 4 is 2.

Mathematica

f[p_, e_] :=  p^If[Divisible[e, 3], 0, 1]; a[n_] := Times @@ (f @@@ FactorInteger[ n]); Array[a, 100] (* Amiram Eldar, Aug 29 2019 *)

Prog

(PARI) a(n) = my(f=factor(n)); for (k=1, #f~, if (frac(f[k, 2]/3), f[k, 2] = 1, f[k, 2] = 0)); factorback(f); \\ Michel Marcus, Feb 28 2019

Crossrefs

Cf. A000189, A000578, A007947, A008834, A013664, A019555, A048798, A050985, A053149, A053150, A056551.

Sequence in context: A088631 A086184 A363484 * A049273 A053590 A208644

Adjacent sequences: A056549 A056550 A056551 * A056553 A056554 A056555

Keyword

nonn,easy,mult

Author

Henry Bottomley, Jun 25 2000

Status

approved