cp's OEIS Frontend

A239728 Perfect power but neither square nor cube.

%I A239728 #40 Aug 23 2025 09:29:47
%S A239728 32,128,243,2048,2187,3125,7776,8192,16807,78125,100000,131072,161051,
%T A239728 177147,248832,279936,371293,524288,537824,759375,823543,1419857,
%U A239728 1594323,1889568,2476099,3200000,4084101,5153632,6436343,7962624,8388608,10000000,11881376,17210368
%N A239728 Perfect power but neither square nor cube.
%H A239728 Reinhard Zumkeller, <a href="/A239728/b239728.txt">Table of n, a(n) for n = 1..1000</a>
%F A239728 GCD(A052409(a(n)), 6) = 1. - _Reinhard Zumkeller_, Mar 28 2014
%F A239728 Sum_{n>=1} 1/a(n) = 1 - zeta(2) - zeta(3) + zeta(6) + Sum_{k>=2} mu(k)*(1-zeta(k)) = 0.0448164603... - _Amiram Eldar_, Dec 21 2020
%e A239728 279936 is included since 279936 = 6^7 is a power and this is not a square or a cube.
%e A239728 59049 = 9^5 not included since this is a square 243^2 = 59049.
%e A239728 32768 = 8^5 not included since this is a cube 32^3 = 32768.
%o A239728 (PARI) for(i=1, 2^25, if(gcd(ispower(i), 6) == 1, print(i)))
%o A239728 (Python)
%o A239728 from sympy import mobius, integer_nthroot
%o A239728 def A239728(n):
%o A239728     def f(x): return int(n+x-integer_nthroot(x,5)[0]+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(7,x.bit_length())))
%o A239728     kmin, kmax = 1,2
%o A239728     while f(kmax) >= kmax:
%o A239728         kmax <<= 1
%o A239728     while True:
%o A239728         kmid = kmax+kmin>>1
%o A239728         if f(kmid) < kmid:
%o A239728             kmax = kmid
%o A239728         else:
%o A239728             kmin = kmid
%o A239728         if kmax-kmin <= 1:
%o A239728             break
%o A239728     return kmax # _Chai Wah Wu_, Aug 14 2024
%Y A239728 Cf. A001597 (perfect powers), A097054 (nonsquare perfect powers), A340585 (noncube perfect powers).
%Y A239728 Cf. A008683, A052409.
%K A239728 nonn,easy,changed
%O A239728 1,1
%A A239728 _Jeppe Stig Nielsen_, Mar 25 2014