A056551 Smallest cube divisible by n divided by largest cube which divides n.

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

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Henry Bottomley, Some Smarandache-type multiplicative sequences.

Formula

a(n) = A053149(n)/A008834(n) = A048798(n)*A050985(n) = A056552(n)^3.

From Amiram Eldar, Oct 28 2022: (Start)

Multiplicative with a(p^e) = 1 if e is divisible by 3, and a(p^e) = p^3 otherwise.

Sum_{k=1..n} a(k) ~ c * n^4, where c = (zeta(12)/(4*zeta(3))) * Product_{p prime} (1 - 1/p^2 + 1/p^3) = A013670 * A330596 / (4*A002117) = 0.1557163105... . (End)

Dirichlet g.f.: zeta(3*s) * Product_{p prime} (1 + 1/p^(s-3) + 1/p^(2*s-3)). - Amiram Eldar, Sep 16 2023

Example

a(16) = 8 since smallest cube divisible by 16 is 64 and smallest cube which divides 16 is 8 and 64/8 = 8.

Mathematica

f[p_, e_] := p^If[Divisible[e, 3], 0, 1]; a[n_] := (Times @@ (f @@@ FactorInteger[ n]))^3; Array[a, 100] (* Amiram Eldar, Aug 29 2019*)

Prog

(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2]%3, f[i, 1], 1))^3; } \\ Amiram Eldar, Oct 28 2022

Crossrefs

Cf. A000189, A000578, A008834, A019555, A048798, A050985, A053149, A053150, A056552.

Cf. A002117, A013670, A330596.

Sequence in context: A070490 A088378 A005064 * A356193 A356192 A367934

Adjacent sequences: A056548 A056549 A056550 * A056552 A056553 A056554

Keyword

nonn,easy,mult

Author

Henry Bottomley, Jun 25 2000

Status

approved