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A076433 Perfect powers for which the two closest perfect powers are greater.

%I A076433 #18 Sep 10 2024 00:24:15
%S A076433 25,121,2187,6431296,6434856,6956883693,27027009001,34359738368,
%T A076433 42618264157,312078649600,312079600999,328080365089,11305780833649,
%U A076433 11305786504384,19287643015432,62854896459664,79723523012809
%N A076433 Perfect powers for which the two closest perfect powers are greater.
%H A076433 Chai Wah Wu, <a href="/A076433/b076433.txt">Table of n, a(n) for n = 1..22</a>
%e A076433 The two closest perfect powers to 25 are 27 (difference = 2) and 32 (difference = 7). The third closest is 16 (difference = 9). Both 27 and 32 are greater than 25, so 25 is in the list.
%o A076433 (Python)
%o A076433 from itertools import count, islice
%o A076433 from sympy import mobius, integer_nthroot
%o A076433 def A076433_gen(): # generator of terms
%o A076433     def f(x): return int(x+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
%o A076433     def bisection(f,kmin=0,kmax=1):
%o A076433         while f(kmax) > kmax: kmax <<= 1
%o A076433         while kmax-kmin > 1:
%o A076433             kmid = kmax+kmin>>1
%o A076433             if f(kmid) <= kmid:
%o A076433                 kmax = kmid
%o A076433             else:
%o A076433                 kmin = kmid
%o A076433         return kmax
%o A076433     a = bisection(f)
%o A076433     b = bisection(lambda x:f(x)+1,a,a)
%o A076433     c = bisection(lambda x:f(x)+2,b,b)
%o A076433     for i in count(3):
%o A076433         d = bisection(lambda x:f(x)+i,c,c)
%o A076433         if b-a > d-b:
%o A076433             yield b
%o A076433         a,b,c=b,c,d
%o A076433 A076433_list = list(islice(A076433_gen(),5)) # _Chai Wah Wu_, Sep 09 2024
%Y A076433 Cf. A001597, A053289, A075772, A076431, A076432.
%K A076433 nonn
%O A076433 1,1
%A A076433 _Neil Fernandez_, Oct 10 2002
%E A076433 More terms from _Jud McCranie_ and _Robert G. Wilson v_, Oct 11 2002
%E A076433 a(6)-a(17) from _Donovan Johnson_, Sep 03 2008