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A053149 Smallest cube divisible by n.

%I A053149 #34 Oct 27 2022 07:31:37
%S A053149 1,8,27,8,125,216,343,8,27,1000,1331,216,2197,2744,3375,64,4913,216,
%T A053149 6859,1000,9261,10648,12167,216,125,17576,27,2744,24389,27000,29791,
%U A053149 64,35937,39304,42875,216,50653,54872,59319,1000,68921,74088,79507,10648
%N A053149 Smallest cube divisible by n.
%H A053149 Amiram Eldar, <a href="/A053149/b053149.txt">Table of n, a(n) for n = 1..10000</a>
%H A053149 Henry Bottomley, <a href="http://fs.gallup.unm.edu/Bottomley-Sm-Mult-Functions.htm">Some Smarandache-type multiplicative sequences</a>.
%F A053149 a(n) = (n/A000189(n))^3 = A008834(n)*A019554(A050985(n))^3 = n*A050985(n)^2/A000188(A050985(n))^3.
%F A053149 a(n) = n * A048798(n). - _Franklin T. Adams-Watters_, Apr 08 2009
%F A053149 From _Amiram Eldar_, Jul 29 2022: (Start)
%F A053149 Multiplicative with a(p^e) = p^(e + ((3-e) mod 3)).
%F A053149 Sum_{n>=1} 1/a(n) = Product_{p prime} ((p^3+2)/(p^3-1)) = 1.655234386560802506... . (End)
%F A053149 Sum_{k=1..n} a(k) ~ c * n^4, where c = (zeta(9)/(4*zeta(3))) * Product_{p prime} (1 - 1/p^2 + 1/p^3) = A013667*A330596/(4*A002117) = 0.1559906... . - _Amiram Eldar_, Oct 27 2022
%t A053149 a[n_] := For[k = 1, True, k++, If[ Divisible[c = k^3, n], Return[c]]]; Table[a[n], {n, 1, 44}] (* _Jean-François Alcover_, Sep 03 2012 *)
%t A053149 f[p_, e_] := p^(e + Mod[3 - e, 3]); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Aug 29 2019 *)
%t A053149 scdn[n_]:=Module[{c=Ceiling[Surd[n,3]]},While[!Divisible[c^3,n],c++];c^3]; Array[scdn,50] (* _Harvey P. Dale_, Jun 13 2020 *)
%o A053149 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^(f[i,2] + (3-f[i,2])%3));} \\ _Amiram Eldar_, Oct 27 2022
%Y A053149 Cf. A000188, A000189, A007913, A008834, A019554, A048798, A050985.
%Y A053149 Cf. A002117, A013667, A330596.
%K A053149 nonn,easy,mult
%O A053149 1,2
%A A053149 _Henry Bottomley_, Feb 28 2000