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A074451 Non-cubefree noncubes.

%I A074451 #25 Jul 27 2025 08:56:31
%S A074451 16,24,32,40,48,54,56,72,80,81,88,96,104,108,112,120,128,135,136,144,
%T A074451 152,160,162,168,176,184,189,192,200,208,224,232,240,243,248,250,256,
%U A074451 264,270,272,280,288,296,297,304,312,320,324,328,336,344,351,352,360
%N A074451 Non-cubefree noncubes.
%H A074451 Amiram Eldar, <a href="/A074451/b074451.txt">Table of n, a(n) for n = 1..10000</a>
%F A074451 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]
%F A074451 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
%t A074451 With[{m = 10}, Select[Complement[Range[m^3], Range[m]^3], AnyTrue[FactorInteger[#][[;; , 2]], #1 > 2 &] &]] (* _Amiram Eldar_, Aug 31 2024 *)
%o A074451 (PARI) is(n)=my(f=factor(n)[,2]); f%3 && vecmax(f)>2 \\ _Charles R Greathouse IV_, Oct 16 2015
%o A074451 (Python)
%o A074451 from sympy import integer_nthroot, mobius
%o A074451 def A074451(n):
%o A074451     def bisection(f,kmin=0,kmax=1):
%o A074451         while f(kmax) > kmax: kmax <<= 1
%o A074451         kmin = kmax >> 1
%o A074451         while kmax-kmin > 1:
%o A074451             kmid = kmax+kmin>>1
%o A074451             if f(kmid) <= kmid:
%o A074451                 kmax = kmid
%o A074451             else:
%o A074451                 kmin = kmid
%o A074451         return kmax
%o A074451     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))
%o A074451     return bisection(f,n,n) # _Chai Wah Wu_, Jun 05 2025
%Y A074451 Intersection of A046099 and A007412.
%Y A074451 Cf. A051144, A002117.
%K A074451 nonn
%O A074451 1,1
%A A074451 _Reinhard Zumkeller_, Sep 25 2002