A360539 a(n) is the cubefree part of n: the largest unitary divisor of n that is a cubefree number (A004709).

1, 2, 3, 4, 5, 6, 7, 1, 9, 10, 11, 12, 13, 14, 15, 1, 17, 18, 19, 20, 21, 22, 23, 3, 25, 26, 1, 28, 29, 30, 31, 1, 33, 34, 35, 36, 37, 38, 39, 5, 41, 42, 43, 44, 45, 46, 47, 3, 49, 50, 51, 52, 53, 2, 55, 7, 57, 58, 59, 60, 61, 62, 63, 1, 65, 66, 67, 68, 69, 70

Offset: 1

Author

Amiram Eldar, Feb 11 2023

Keywords

• nonn
• easy
• mult

Comments

Equivalently, a(n) is the least divisor d of n such that n/d is a cubefull number (A036966).

Links

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

Crossrefs

Cf. A004709, A007948, A036966, A055231, A360540.

Programs

Mathematica

f[p_, e_] := If[e < 3, p^e, 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, f[i, 1]^f[i, 2], 1));}

Python

from math import prod
from sympy import factorint
def A360539(n): return prod(p**e for p,e in factorint(n).items() if e<=2) # Chai Wah Wu, Aug 06 2024

Formula

a(n) = 1 if and only if n is a cubefull number (A036966).

a(n) = n if and only if n is a cubefree number (A004709).

a(n) >= A055231(n) with equality if and only if n is in A337050.

a(n) = n/A360540(n).

Multiplicative with a(p^e) = p^e if e <= 2, and 1 otherwise.

Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + p^(1-s) - p^(-s) + p^(2-2*s) - p^(1-2*s) - p^(2-3*s) + p^(-3*s)).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^5 + 1/p^6 - 1/p^7) = 0.4213813264... .