A362147 -id:A362147 - cp's OEIS frontend

A062838 Cubes of squarefree numbers.

1, 8, 27, 125, 216, 343, 1000, 1331, 2197, 2744, 3375, 4913, 6859, 9261, 10648, 12167, 17576, 24389, 27000, 29791, 35937, 39304, 42875, 50653, 54872, 59319, 68921, 74088, 79507, 97336, 103823, 132651, 148877, 166375, 185193, 195112, 205379, 226981, 238328

Offset: 1

Jason Earls, Jul 21 2001

Cubefull numbers (A036966) all of whose nonunitary divisors are not cubefull (A362147). - Amiram Eldar, May 13 2023

Vincenzo Librandi and T. D. Noe, Table of n, a(n) for n = 1..1000

Other powers of squarefree numbers: A005117(1), A062503(2), A113849(4), A072774(all).
Cf. A000400, A036966, A225546, A362147.
A329332 column 3 in ascending order.

Select[Range[70], SquareFreeQ]^3 (* Harvey P. Dale, Jul 20 2011 *)

for(n=1,35, if(issquarefree(n),print(n*n^2)))

a(n) = my(m, c); if(n<=1, n==1, c=1; m=1; while(cAltug Alkan, Dec 03 2015

from math import isqrt
from sympy import mobius
def A062838(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m**3 # Chai Wah Wu, Sep 11 2024

A055229(a(n)) = A005117(n) and A055229(m) < A005117(n) for m < a(n). - Reinhard Zumkeller, Apr 09 2010
a(n) = A005117(n)^3. - R. J. Mathar, Dec 03 2015
{a(n)} = {A225546(A000400(n)) : n >= 0}, where {a(n)} denotes the set of integers in the sequence. - Peter Munn, Oct 31 2019
Sum_{n>=1} 1/a(n) = zeta(3)/zeta(6) = 945*zeta(3)/Pi^6 (A157289). - Amiram Eldar, May 22 2020

More terms from Dean Hickerson, Jul 24 2001

More terms from Vladimir Joseph Stephan Orlovsky, Aug 15 2008

A362148 Numbers that are neither cubefree nor cubefull.

Original entry on oeis.org

24, 40, 48, 54, 56, 72, 80, 88, 96, 104, 108, 112, 120, 135, 136, 144, 152, 160, 162, 168, 176, 184, 189, 192, 200, 208, 224, 232, 240, 248, 250, 264, 270, 272, 280, 288, 296, 297, 304, 312, 320, 324, 328, 336, 344, 351, 352, 360, 368, 375, 376, 378, 384, 392, 400

Offset: 1

Bernard Schott, Apr 09 2023

In fact, every cubefull number > 1 is noncubefree, but the converse is false.
This sequence = A046099 \ A036966 and lists these counterexamples.
Numbers k such that for some primes p and q, k is divisible by p^3*q but not by q^3. - Robert Israel, Apr 28 2023
The asymptotic density of this sequence is 1 - 1/zeta(3) = 0.168092... - Charles R Greathouse IV, Apr 28 2023
From Amiram Eldar, Sep 17 2023: (Start)
Numbers k such that A360539(k) > 1 and A360540(k) > 1.
Equivalently, numbers that have in their prime factorization at least one exponent that is smaller than 3 and at least one exponent that is larger than 2. (End)

			24 = 2^3 * 3 is noncubefree as it is divisible by the cube 2^3, but it is not cubefull because 3 divides 24 but 3^3 does not divide 24, hence 24 is a term.
648 = 2^4 * 3^3 is noncubefree as it is divisible by the cube 3^3, but it is also cubefull because primes 2 and 3 divide 648, and 2^3 and 3^3 divide also 648, so 648 is not a term.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Intersection of A046099 (not cubefree) and A362147 (not cubefull)

Cf. A004709 (cubefree), A036966 (cubefull), A360539, A360540.

filter:= proc(n) local F;
F:= ifactors(n)[2][..,2];
  min(F) < 3 and max(F) >= 3
end proc:
select(filter, [$1..400]); # Robert Israel, Apr 28 2023

Select[Range[500], Min[(e = FactorInteger[#][[;; , 2]])] < 3 && Max[e] > 2 &] (* Amiram Eldar, Apr 09 2023 *)

isok(k) = (k>1) && (vecmax(factor(k)[, 2])>2) && (vecmin(factor(k)[, 2])<=2); \\ Michel Marcus, Apr 19 2023

Equals A362147 \ A004709.

Sum_{n>=1} 1/a(n) = 1 + zeta(s) - zeta(s)/zeta(3*s) - Product_{p prime}(1 + 1/(p^(2*s)*(p^s-1))), s > 1. - Amiram Eldar, Sep 17 2023

A378287 Numbers not of the form m^k for some k>=3. Complement of A076467.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73

Offset: 1

Chai Wah Wu, Nov 21 2024

Differs from A362147 at a(419).

Cf. A076467, A362147.

from sympy import integer_nthroot, mobius
def A378287(n):
    def f(x): return int(n+integer_nthroot(x,4)[0]-sum(mobius(k)*(integer_nthroot(x,k)[0]+integer_nthroot(x,k<<1)[0]-2) for k in range(3,x.bit_length())))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m

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A062838 Cubes of squarefree numbers.

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A362148 Numbers that are neither cubefree nor cubefull.

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A378287 Numbers not of the form m^k for some k>=3. Complement of A076467.

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Python