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A212793 Characteristic function of cubefree numbers, A004709. 29

%I

%S 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,

%T 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,

%U 1,1,1,0,1,1,1,1,1,1,1,0,0,1,1,1,1,1

%N Characteristic function of cubefree numbers, A004709.

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

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

%C 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

%H Reinhard Zumkeller, <a href="/A212793/b212793.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Cubefree.html">Cubefree</a>

%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>

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

%F Multiplicative with a(p^e) = 1 if e<=2, =0 if e>=3. - _R. J. Mathar_, Dec 17 2012

%F sum(n>0, a(n)/n^s) = product(p prime, 1+p^(-s)+p^(-2s) ) = zeta(s) / zeta(3s). - _Ralf Stephan_, Jul 07 2013

%t Table[Boole[Max[FactorInteger[n][[All, 2]]] < 3], {n, 1, 100}] (* _Geoffrey Critzer_, Feb 25 2015 *)

%o (Haskell)

%o a212793 = cubeFree a000040_list 0 0 where

%o cubeFree ps'@(p:ps) q e x

%o | e > 2 = 0

%o | x == 1 = 1

%o | r > 0 = cubeFree ps p 0 x

%o | otherwise = cubeFree ps' p (e + 1) x' where (x', r) = divMod x p

%o -- _Reinhard Zumkeller_, Mar 04 2015, May 27 2012

%o (PARI) a(n) = {f = factor(n); for (i=1, #f~, if ((f[i,2]) >=3, return(0));); return (1);} \\ _Michel Marcus_, Feb 10 2015

%Y Cf. A000005, A000007, A004709, A008966, A046099, A053864, A060431, A124010.

%K nonn,mult

%O 1

%A _Reinhard Zumkeller_, May 27 2012

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Last modified December 15 17:03 EST 2019. Contains 330000 sequences. (Running on oeis4.)