%I #78 Aug 05 2024 16:03:54
%S 8,16,24,27,32,40,48,54,56,64,72,80,81,88,96,104,108,112,120,125,128,
%T 135,136,144,152,160,162,168,176,184,189,192,200,208,216,224,232,240,
%U 243,248,250,256,264,270,272,280,288,296,297,304,312,320,324,328,336
%N Numbers that are not cubefree. Numbers divisible by a cube greater than 1. Complement of A004709.
%C Also called cubeful numbers, but this term is ambiguous and is best avoided.
%C Numbers n such that A007427(n) = sum(d|n,mu(d)*mu(n/d)) == 0. - _Benoit Cloitre_, Apr 17 2002
%C The convention in the OEIS is that squareful, cubeful, biquadrateful (A046101), ... mean the same as "not squarefree" etc., while 2- or square-full, 3- or cube-full (A036966), 4-full (A036967) are used for Golomb's notion of powerful numbers (A001694, see references there), when each prime factor occurs to a power > 1. - _M. F. Hasler_, Feb 12 2008. Added by _N. J. A. Sloane_, Apr 25 2023: This suggestion has not been a success. It is hopeless to try to make a distinction between "cubeful" and "cubefull". To avoid ambiguity, do not use either term, but instead say exactly what you mean.
%C Also solutions to equation tau_{-2}(n)=0, where tau_{-2} is A007427. - _Enrique PĂ©rez Herrero_, Jan 19 2013
%C The asymptotic density of this sequence is 1 - 1/zeta(3) = 0.168092... - _Amiram Eldar_, Jul 09 2020
%H T. D. Noe, <a href="/A046099/b046099.txt">Table of n, a(n) for n = 1..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Cubefree.html">Cubefree</a>.
%F A212793(a(n)) = 0. - _Reinhard Zumkeller_, May 27 2012
%F Sum_{n>=1} 1/a(n)^s = (zeta(s)*(zeta(3*s)-1))/zeta(3*s). - _Amiram Eldar_, Dec 27 2022
%p isA046099 := proc(n)
%p local p;
%p for p in ifactors(n)[2] do
%p if op(2,p) >= 3 then
%p return true;
%p end if;
%p end do:
%p false ;
%p end proc:
%p for n from 1 do
%p if isA046099(n) then
%p printf("%d\n",n) ;
%p end if;
%p end do: # _R. J. Mathar_, Dec 08 2015
%t lst={};Do[a=0;Do[If[FactorInteger[m][[n, 2]]>2, a=1], {n, Length[FactorInteger[m]]}];If[a==1, AppendTo[lst, m]], {m, 10^3}];lst (* _Vladimir Joseph Stephan Orlovsky_, Aug 15 2008 *)
%o (Haskell)
%o a046099 n = a046099_list !! (n-1)
%o a046099_list = filter ((== 1) . a212793) [1..]
%o -- _Reinhard Zumkeller_, May 27 2012
%o (PARI) is(n)=n>7 && vecmax(factor(n)[,2])>2 \\ _Charles R Greathouse IV_, Sep 17 2015
%o (Python)
%o from sympy.ntheory.factor_ import core
%o def ok(n): return core(n, 3) != n
%o print(list(filter(ok, range(1, 337)))) # _Michael S. Branicky_, Aug 16 2021
%o (Python)
%o from sympy import mobius, integer_nthroot
%o def A046099(n):
%o def f(x): return n+sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))
%o m, k = n, f(n)
%o while m != k:
%o m, k = k, f(k)
%o return m # _Chai Wah Wu_, Aug 05 2024
%Y Complement of A004709.
%Y Subsequences: A000578 and A030078.
%Y Cf. A001694, A036966, A046101, A088453.
%K nonn
%O 1,1
%A _Eric W. Weisstein_
%E More terms from _Vladimir Joseph Stephan Orlovsky_, Aug 15 2008
%E Edited by _N. J. A. Sloane_, Jul 27 2009