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A117453 Perfect powers in more than one way.

%I A117453 #31 Feb 16 2025 08:33:00
%S A117453 1,16,64,81,256,512,625,729,1024,1296,2401,4096,6561,10000,14641,
%T A117453 15625,16384,19683,20736,28561,32768,38416,46656,50625,59049,65536,
%U A117453 83521,104976,117649,130321,160000,194481,234256,262144,279841,331776,390625
%N A117453 Perfect powers in more than one way.
%C A117453 Corresponding values of ways for a(n) in A175066(n) for n >= 2. - _Jaroslav Krizek_, Jan 23 2010
%C A117453 Perfect powers expressible as m^k with k composite. - _Charlie Neder_, Mar 02 2019
%H A117453 Donovan Johnson, <a href="/A117453/b117453.txt">Table of n, a(n) for n = 1..10000</a>
%H A117453 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PerfectPower.html">Perfect Power</a>
%e A117453 16 = 2^4 = 4^2.
%t A117453 s = Split@ Sort@ Flatten@ Table[ n^i, {n, 2, Sqrt@456975}, {i, 2, Log[n, 456975]}]; Union@ Flatten@ Select[s, Length@ # > 1 &] (* _Robert G. Wilson v_, Apr 12 2006 *)
%o A117453 (Python)
%o A117453 from sympy import mobius, integer_nthroot, primerange
%o A117453 def A117453(n):
%o A117453     def bisection(f,kmin=0,kmax=1):
%o A117453         while f(kmax) > kmax: kmax <<= 1
%o A117453         while kmax-kmin > 1:
%o A117453             kmid = kmax+kmin>>1
%o A117453             if f(kmid) <= kmid:
%o A117453                 kmax = kmid
%o A117453             else:
%o A117453                 kmin = kmid
%o A117453         return kmax
%o A117453     def f(x): return int(n+sum(mobius(k)*(integer_nthroot(x,k)[0]-1+sum(integer_nthroot(x,p*k)[0]-1 for p in primerange((x//k).bit_length()))) for k in range(1,x.bit_length())))
%o A117453     return bisection(f,n,n) # _Chai Wah Wu_, Nov 24 2024
%Y A117453 Cf. A001597, A322449.
%K A117453 nonn
%O A117453 1,2
%A A117453 _Eric W. Weisstein_, Mar 16 2006
%E A117453 More terms from _Robert G. Wilson v_, Apr 12 2006