A366126 The largest unitary divisor of n that is a cube.

1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 27, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 27, 1, 8, 1, 1, 1, 1, 1, 1, 1, 64, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Amiram Eldar, Sep 30 2023

Keywords

• nonn
• easy
• mult

Comments

First differs from A056191 at n = 32.

Links

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

Crossrefs

Cf. A001221, A001222, A000578, A056191, A350388, A366124, A366125.

Programs

Mathematica

f[p_, e_] := If[Divisible[e, 3], p^e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] % 3, 1, f[i,1]^f[i,2]));}

Formula

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

a(n) = n if and only if n is a positive cube (A000578 \ {0}).

A001221(a(n)) = A366124(n).

A001222(a(n)) = A366125(n).

Sum_{k=1..n} a(k) ~ c * n^(4/3), where c = (1/4) * Product_{p prime} (1 + (p^(1/3) + p^(5/3))/(1 + p + p^2 + p^3)) = 0.61488587249270755696... .