OFFSET
1,1
COMMENTS
Let lp(n, e) denote the largest positive integer b such that b^e divides n. For example for e = 1, 2, 3, 4 the sequences (lp(n, e), n >= 1) are A000027, A000188, A053150, and A053164. Let rad(n) = A007947(n) be the squarefree kernel of n. k is in this sequence if lp(n, 3) does not divide rad(n). The case e = 1 gives A013929, and the case e = 2 is A046101.
The asymptotic density of this sequence is 1 - 1/zeta(6) = 1 - 945/Pi^6 = 0.017047... . - Amiram Eldar, Jul 13 2022
A number is bicubeful iff it is divisible by the 6th power of an integer > 1
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
EXAMPLE
n = 512 = 2^9, rad(n) = 2, lp(n, 3) = 8 since n/8^3 = 1. But 8 does not divide 2.
n = 704 = 2^6*11, rad(n) = 22, lp(n, 3) = 4 since n/4^3 = 11. But 4 does not divide 22.
MAPLE
with(NumberTheory):
isBicubeful := n -> irem(Radical(n), LargestNthPower(n, 3)) <> 0:
select(isBicubeful, [`$`(1..2752)]);
MATHEMATICA
bicubQ[n_] := AnyTrue[FactorInteger[n][[;; , 2]], # > 5 &]; Select[Range[3000], bicubQ] (* Amiram Eldar, Jul 13 2022 *)
Select[Range[3000], Max[FactorInteger[#][[;; , 2]]]>5&] (* Harvey P. Dale, Jun 05 2026 *)
PROG
(Python)
from itertools import count, islice
from sympy import factorint
def A355265_gen(startvalue=1): # generator of terms >= startvalue
return filter(lambda n:any(map(lambda m:m>5, factorint(n).values())), count(max(startvalue, 1)))
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Peter Luschny, Jul 12 2022
EXTENSIONS
Previous entry in Formula field slightly revised and moved to Comment field by Harvey P. Dale, Jun 05 2026
STATUS
approved