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 A004709 Cubefree numbers: numbers that are not divisible by any cube > 1. 168
 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Numbers n such that no smaller number m satisfies: kronecker(n,k)=kronecker(m,k) for all k. - Michael Somos, Sep 22 2005 The asymptotic density of cubefree integers is the reciprocal of Apery's constant 1/zeta(3) = A088453. - Gerard P. Michon, May 06 2009 The Schnirelmann density of the cubefree numbers is 157/189 (Orr, 1969). - Amiram Eldar, Mar 12 2021 From Amiram Eldar, Feb 26 2024: (Start) Numbers whose sets of unitary divisors (A077610) and bi-unitary divisors (A222266) coincide. Number whose all divisors are (1+e)-divisors, or equivalently, numbers k such that A049599(k) = A000005(k). (End) LINKS Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe) Gérard P. Michon, On the number of cubefree integers not exceeding N. Richard C. Orr, On the Schnirelmann density of the sequence of k-free integers, Journal of the London Mathematical Society, Vol. 1, No. 1 (1969), pp. 313-319. Vladimir Shevelev, Set of all densities of exponentially S-numbers, arXiv preprint, arXiv:1511.03860 [math.NT], 2015. Eric Weisstein's World of Mathematics, Cubefree. FORMULA A066990(a(n)) = a(n). - Reinhard Zumkeller, Jun 25 2009 A212793(a(n)) = 1. - Reinhard Zumkeller, May 27 2012 A124010(a(n),k) <= 2 for all k = 1..A001221(a(n)). - Reinhard Zumkeller, Mar 04 2015 Sum_{n>=1} 1/a(n)^s = zeta(s)/zeta(3*s), for s > 1. - Amiram Eldar, Dec 27 2022 MAPLE isA004709 := proc(n) local p; for p in ifactors(n)[2] do if op(2, p) > 2 then return false; end if; end do: true ; end proc: MATHEMATICA Select[Range[6!], FreeQ[FactorInteger[#], {_, k_ /; k > 2}] &] (* Jan Mangaldan, May 07 2014 *) PROG (PARI) {a(n)= local(m, c); if(n<2, n==1, c=1; m=1; while( cvecmax(factor(m)[, 2]), c++)); m)} /* Michael Somos, Sep 22 2005 */ (Haskell) a004709 n = a004709_list !! (n-1) a004709_list = filter ((== 1) . a212793) [1..] -- Reinhard Zumkeller, May 27 2012 (Python) from sympy.ntheory.factor_ import core def ok(n): return core(n, 3) == n print(list(filter(ok, range(1, 86)))) # Michael S. Branicky, Aug 16 2021 (Python) from sympy import mobius, integer_nthroot def A004709(n): def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x, 3)[0]+1)) m, k = n, f(n) while m != k: m, k = k, f(k) return m # Chai Wah Wu, Aug 05 2024 CROSSREFS Complement of A046099. Cf. A005117 (squarefree), A067259 (cubefree but not squarefree), A046099 (cubeful). Cf. A160112, A160113, A160114 & A160115: On the number of cubefree integers. - Gerard P. Michon, May 06 2009 Cf. A030078. Cf. A001221, A124010, A212793. Cf. A000005, A049599, A077610, A222266. Sequence in context: A023802 A007915 A344742 * A048107 A342521 A078129 Adjacent sequences: A004706 A004707 A004708 * A004710 A004711 A004712 KEYWORD nonn,easy AUTHOR Steven Finch, Jun 14 1998 STATUS approved

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Last modified September 17 10:45 EDT 2024. Contains 375987 sequences. (Running on oeis4.)