|
%I #57 Dec 27 2022 02:28:23
%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 1,1,1,0,1,1,1,1,1,1,1,0,0,1,1,1,1,1,1,0,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,0,1
%N Characteristic function of cubefree numbers, A004709.
%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 Antti Karttunen, <a href="/A212793/b212793.txt">Table of n, a(n) for n = 1..100000</a> (first 10000 terms from Reinhard Zumkeller)
%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(A004709(n)) = 1, a(A046099(n)) = 0;
%F a(n) = A000007(A000005(n) - A073184(n)).
%F a(n) = abs(A053864(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
%F a(n) = Sum_{d|n} A008966(n/d) * A307423(d). - _Antti Karttunen_, Jul 14 2022
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/zeta(3) (A088453). - _Amiram Eldar_, Jul 23 2022
%F Dirichlet g.f.: zeta(s)/zeta(3*s). - _Amiram Eldar_, Dec 27 2022
%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
%o (PARI) A212793(n) = factorback(apply(e->(e<=2), factor(n)[,2])); \\ _Antti Karttunen_, Jul 14 2022
%Y Cf. A000005, A000007, A004709, A008966, A046099, A053864, A060431, A088453, A112526, A124010, A307423.
%K nonn,mult
%O 1
%A _Reinhard Zumkeller_, May 27 2012
%E Data section extended up to a(105) by _Antti Karttunen_, Jul 14 2022
|
