A050985 Cubefree part of n.

1, 2, 3, 4, 5, 6, 7, 1, 9, 10, 11, 12, 13, 14, 15, 2, 17, 18, 19, 20, 21, 22, 23, 3, 25, 26, 1, 28, 29, 30, 31, 4, 33, 34, 35, 36, 37, 38, 39, 5, 41, 42, 43, 44, 45, 46, 47, 6, 49, 50, 51, 52, 53, 2, 55, 7, 57, 58, 59, 60, 61, 62, 63, 1, 65, 66, 67, 68, 69, 70, 71, 9, 73, 74, 75

Offset: 1

Author

Eric W. Weisstein, Dec 11 1999

Keywords

• nonn
• easy
• mult

Comments

This is an unusual sequence in the sense that the 83.2% of the integers that belong to A004709 occur infinitely many times, whereas the remaining 16.8% of the integers that belong to A046099 never occur at all. - Ant King, Sep 22 2013

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
• Henry Bottomley, Some Smarandache-type multiplicative sequences.
• Eric Weisstein's World of Mathematics, Cubefree Part.
• Eric Weisstein's World of Mathematics, Dirichlet Generating Function.

Crossrefs

Cf. A007913, A008834, A053165, A004709, A046099, A301596, A301597.

Programs

Maple

A050985 := proc(n)
    n/A008834(n) ;
end proc:
seq(A050985(n),n=1..40) ; # R. J. Mathar, Dec 08 2015

Mathematica

cf[n_]:=Module[{tr=Transpose[FactorInteger[n]],ex,cb},ex= tr[[2]]- Mod[ tr[[2]],3];cb=Times@@(First[#]^Last[#]&/@Transpose[{tr[[1]], ex}]);n/cb]; Array[cf,75] (* Harvey P. Dale, Jun 03 2012 *)
f[p_, e_] := p^Mod[e, 3]; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 07 2020 *)

PARI

a(n) = my(f=factor(n)); f[,2] = apply(x->(x % 3), f[,2]); factorback(f); \\ Michel Marcus, Jan 06 2019

Python

from operator import mul
from functools import reduce
from sympy import factorint
def A050985(n):
    return 1 if n <=1 else reduce(mul,[p**(e % 3) for p,e in factorint(n).items()])
# Chai Wah Wu, Feb 04 2015

Formula

Multiplicative with p^e -> p^(e mod 3), p prime. - Reinhard Zumkeller, Nov 22 2009

Dirichlet g.f.: zeta(3s)*zeta(s-1)/zeta(3s-3). - R. J. Mathar, Feb 11 2011

a(n) = n/A008834(n). - R. J. Mathar, Dec 08 2015

Sum_{k=1..n} a(k) ~ Pi^6 * n^2 / (1890*Zeta(3)). - Vaclav Kotesovec, Feb 08 2019