A076470 - OEIS

%I #17 Aug 14 2024 13:11:58

%S 1,64,128,256,512,729,1024,2048,2187,4096,6561,8192,15625,16384,19683,

%T 32768,46656,59049,65536,78125,117649,131072,177147,262144,279936,

%U 390625,524288,531441,823543,1000000,1048576,1594323,1679616,1771561

%N Perfect powers m^k where k >= 6.

%C A necessary but not sufficient condition is that if p|n when at least p^6|n.

%H Amiram Eldar, <a href="/A076470/b076470.txt">Table of n, a(n) for n = 1..10000</a>

%F Sum_{n>=1} 1/a(n) = 5 - zeta(2) - zeta(3) - zeta(4) - zeta(5) + Sum_{k>=2} mu(k)*(5 - zeta(k) - zeta(2*k) - zeta(3*k) - zeta(4*k) - zeta(5*k)) = 1.03342597171... . - _Amiram Eldar_, Dec 03 2022

%t a = {1}; Do[ If[ Apply[ GCD, Last[ Transpose[ FactorInteger[n]]]] > 4, a = Append[a, n]; Print[n]], {n, 2, 1953124}]; a

%o (Python)

%o from sympy import mobius, integer_nthroot

%o def A076470(n):

%o     def f(x): return int(n+9+x-(sum(integer_nthroot(x,d)[0] for d in (6,10,15))<<1)-sum(integer_nthroot(x,d)[0] for d in (8,9,12,20,25))+sum(mobius(k)*(sum(integer_nthroot(x,k*i)[0] for i in range(1,6))-5) for k in range(6,x.bit_length())))

%o     kmin, kmax = 1,2

%o     while f(kmax) >= kmax:

%o         kmax <<= 1

%o     while True:

%o         kmid = kmax+kmin>>1

%o         if f(kmid) < kmid:

%o             kmax = kmid

%o         else:

%o             kmin = kmid

%o         if kmax-kmin <= 1:

%o             break

%o     return kmax # _Chai Wah Wu_, Aug 14 2024

%Y Cf. A001597, A076467, A076468, A076469.

%Y Different from A069493.

%Y Cf. A002117, A013661, A013662, A013663.

%K nonn

%O 1,2

%A _Robert G. Wilson v_, Oct 14 2002