A212793 Characteristic function of cubefree numbers, A004709.

1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1

Offset

1

Comments

The following four statements are equivalent: m is cubefree; a(m) = 1; m = A004709(k) for some k; A124010(m,k) <= 2 for all k = 1..A001221(m). - Reinhard Zumkeller, Mar 04 2015

Links

Antti Karttunen, Table of n, a(n) for n = 1..100000 (first 10000 terms from Reinhard Zumkeller)

Eric Weisstein's World of Mathematics, Cubefree.

Index entries for characteristic functions.

Formula

a(A004709(n)) = 1, a(A046099(n)) = 0;

a(n) = A000007(A000005(n) - A073184(n)).

a(n) = abs(A053864(n)).

Multiplicative with a(p^e) = 1 if e<=2, =0 if e>=3. - R. J. Mathar, Dec 17 2012

Sum_{n>0} a(n)/n^s = Product_{p prime} (1+p^(-s)+p^(-2s)) = zeta(s) / zeta(3s). - Ralf Stephan, Jul 07 2013

a(n) = Sum_{d|n} A008966(n/d) * A307423(d). - Antti Karttunen, Jul 14 2022

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/zeta(3) (A088453). - Amiram Eldar, Jul 23 2022

Dirichlet g.f.: zeta(s)/zeta(3*s). - Amiram Eldar, Dec 27 2022

Mathematica

Table[Boole[Max[FactorInteger[n][[All, 2]]] < 3], {n, 1, 100}] (* Geoffrey Critzer, Feb 25 2015 *)

Prog

(Haskell)
a212793 = cubeFree a000040_list 0 0 where
   cubeFree ps'@(p:ps) q e x
      | e > 2     = 0
      | x == 1    = 1
      | r > 0     = cubeFree ps  p 0 x
      | otherwise = cubeFree ps' p (e + 1) x' where (x', r) = divMod x p
-- Reinhard Zumkeller, Mar 04 2015, May 27 2012
(PARI) a(n) = {f = factor(n); for (i=1, #f~, if ((f[i, 2]) >=3, return(0)); ); return (1); } \\ Michel Marcus, Feb 10 2015
(PARI) A212793(n) = factorback(apply(e->(e<=2), factor(n)[, 2])); \\ Antti Karttunen, Jul 14 2022

Crossrefs

Cf. A000005, A000007, A004709, A008966, A046099, A053864, A060431, A088453, A112526, A124010, A307423.

Sequence in context: A363551 A053864 A189021 * A307420 A129667 A071374

Adjacent sequences: A212790 A212791 A212792 * A212794 A212795 A212796

Keyword

nonn,mult

Author

Reinhard Zumkeller, May 27 2012

Extensions

Data section extended up to a(105) by Antti Karttunen, Jul 14 2022

Status

approved