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.
FORMULA
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
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
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