A053149 Smallest cube divisible by n.

1, 8, 27, 8, 125, 216, 343, 8, 27, 1000, 1331, 216, 2197, 2744, 3375, 64, 4913, 216, 6859, 1000, 9261, 10648, 12167, 216, 125, 17576, 27, 2744, 24389, 27000, 29791, 64, 35937, 39304, 42875, 216, 50653, 54872, 59319, 1000, 68921, 74088, 79507, 10648

Offset: 1

Author

Henry Bottomley, Feb 28 2000

Keywords

• nonn
• easy
• mult

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Henry Bottomley, Some Smarandache-type multiplicative sequences.

Crossrefs

Cf. A000188, A000189, A007913, A008834, A019554, A048798, A050985.

Cf. A002117, A013667, A330596.

Programs

Mathematica

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 *)
f[p_, e_] := p^(e + Mod[3 - e, 3]); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Aug 29 2019 *)
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 *)

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

Formula

a(n) = (n/A000189(n))^3 = A008834(n)*A019554(A050985(n))^3 = n*A050985(n)^2/A000188(A050985(n))^3.

a(n) = n * A048798(n). - Franklin T. Adams-Watters, Apr 08 2009

From Amiram Eldar, Jul 29 2022: (Start)

Multiplicative with a(p^e) = p^(e + ((3-e) mod 3)).

Sum_{n>=1} 1/a(n) = Product_{p prime} ((p^3+2)/(p^3-1)) = 1.655234386560802506... . (End)

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