A074451 Non-cubefree noncubes.

16, 24, 32, 40, 48, 54, 56, 72, 80, 81, 88, 96, 104, 108, 112, 120, 128, 135, 136, 144, 152, 160, 162, 168, 176, 184, 189, 192, 200, 208, 224, 232, 240, 243, 248, 250, 256, 264, 270, 272, 280, 288, 296, 297, 304, 312, 320, 324, 328, 336, 344, 351, 352, 360

Offset: 1

Author

Reinhard Zumkeller, Sep 25 2002

Keywords

• nonn

Links

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

Crossrefs

Intersection of A046099 and A007412.

Cf. A051144, A002117.

Programs

Mathematica

With[{m = 10}, Select[Complement[Range[m^3], Range[m]^3], AnyTrue[FactorInteger[#][[;; , 2]], #1 > 2 &] &]] (* Amiram Eldar, Aug 31 2024 *)

PARI

is(n)=my(f=factor(n)[,2]); f%3 && vecmax(f)>2 \\ Charles R Greathouse IV, Oct 16 2015

Python

from sympy import integer_nthroot, mobius
def A074451(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n-1+(m:=integer_nthroot(x,3)[0])+sum(mobius(k)*(x//k**3) for k in range(1, m+1))
    return bisection(f,n,n) # Chai Wah Wu, Jun 05 2025

Formula

For n > 35, a(n) < 7n. Asymptotically, a(n) ~ kn with k = zeta(3)/(zeta(3)-1) = 5.949... . - Charles R Greathouse IV, Oct 16 2015 [Corrected by Amiram Eldar, Aug 31 2024]

Sum_{n>=1} 1/a(n)^s = 1 + zeta(s) - zeta(3*s) - zeta(s)/zeta(3*s), for s > 1. - Amiram Eldar, Aug 31 2024