cp's OEIS Frontend

A046099 Numbers that are not cubefree. Numbers divisible by a cube greater than 1. Complement of A004709.

%I A046099 #79 Feb 16 2025 08:32:38
%S A046099 8,16,24,27,32,40,48,54,56,64,72,80,81,88,96,104,108,112,120,125,128,
%T A046099 135,136,144,152,160,162,168,176,184,189,192,200,208,216,224,232,240,
%U A046099 243,248,250,256,264,270,272,280,288,296,297,304,312,320,324,328,336
%N A046099 Numbers that are not cubefree. Numbers divisible by a cube greater than 1. Complement of A004709.
%C A046099 Also called cubeful numbers, but this term is ambiguous and is best avoided.
%C A046099 Numbers n such that A007427(n) = sum(d|n,mu(d)*mu(n/d)) == 0. - _Benoit Cloitre_, Apr 17 2002
%C A046099 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 A046099 Also solutions to equation tau_{-2}(n)=0, where tau_{-2} is A007427. - _Enrique Pérez Herrero_, Jan 19 2013
%C A046099 The asymptotic density of this sequence is 1 - 1/zeta(3) = 0.168092... - _Amiram Eldar_, Jul 09 2020
%H A046099 T. D. Noe, <a href="/A046099/b046099.txt">Table of n, a(n) for n = 1..1000</a>
%H A046099 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Cubefree.html">Cubefree</a>.
%F A046099 A212793(a(n)) = 0. - _Reinhard Zumkeller_, May 27 2012
%F A046099 Sum_{n>=1} 1/a(n)^s = (zeta(s)*(zeta(3*s)-1))/zeta(3*s). - _Amiram Eldar_, Dec 27 2022
%p A046099 isA046099 := proc(n)
%p A046099     local p;
%p A046099     for p in ifactors(n)[2] do
%p A046099         if op(2,p) >= 3 then
%p A046099             return true;
%p A046099         end if;
%p A046099     end do:
%p A046099     false ;
%p A046099 end proc:
%p A046099 for n from 1 do
%p A046099     if isA046099(n) then
%p A046099         printf("%d\n",n) ;
%p A046099     end if;
%p A046099 end do: # _R. J. Mathar_, Dec 08 2015
%t A046099 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 A046099 (Haskell)
%o A046099 a046099 n = a046099_list !! (n-1)
%o A046099 a046099_list = filter ((== 1) . a212793) [1..]
%o A046099 -- _Reinhard Zumkeller_, May 27 2012
%o A046099 (PARI) is(n)=n>7 && vecmax(factor(n)[,2])>2 \\ _Charles R Greathouse IV_, Sep 17 2015
%o A046099 (Python)
%o A046099 from sympy.ntheory.factor_ import core
%o A046099 def ok(n): return core(n, 3) != n
%o A046099 print(list(filter(ok, range(1, 337)))) # _Michael S. Branicky_, Aug 16 2021
%o A046099 (Python)
%o A046099 from sympy import mobius, integer_nthroot
%o A046099 def A046099(n):
%o A046099     def f(x): return n+sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))
%o A046099     m, k = n, f(n)
%o A046099     while m != k:
%o A046099         m, k = k, f(k)
%o A046099     return m # _Chai Wah Wu_, Aug 05 2024
%Y A046099 Complement of A004709.
%Y A046099 Subsequences: A000578 and A030078.
%Y A046099 Cf. A001694, A036966, A046101, A088453.
%K A046099 nonn
%O A046099 1,1
%A A046099 _Eric W. Weisstein_
%E A046099 More terms from _Vladimir Joseph Stephan Orlovsky_, Aug 15 2008
%E A046099 Edited by _N. J. A. Sloane_, Jul 27 2009