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

1,2

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

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... .

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A033928 A194754 A167972 * A319656 A367699 A368171

Adjacent sequences: A360536 A360537 A360538 * A360540 A360541 A360542

Keyword

nonn,easy,mult,changed

Author

Amiram Eldar, Feb 11 2023

Status

approved