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A076433
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Perfect powers for which the two closest perfect powers are greater.
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2
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25, 121, 2187, 6431296, 6434856, 6956883693, 27027009001, 34359738368, 42618264157, 312078649600, 312079600999, 328080365089, 11305780833649, 11305786504384, 19287643015432, 62854896459664, 79723523012809
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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LINKS
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EXAMPLE
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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.
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PROG
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(Python)
from itertools import count, islice
from sympy import mobius, integer_nthroot
def A076433_gen(): # generator of terms
def f(x): return int(x+sum(mobius(k)*(integer_nthroot(x, k)[0]-1) for k in range(2, x.bit_length())))
def bisection(f, kmin=0, kmax=1):
while f(kmax) > kmax: kmax <<= 1
while kmax-kmin > 1:
kmid = kmax+kmin>>1
if f(kmid) <= kmid:
kmax = kmid
else:
kmin = kmid
return kmax
a = bisection(f)
b = bisection(lambda x:f(x)+1, a, a)
c = bisection(lambda x:f(x)+2, b, b)
for i in count(3):
d = bisection(lambda x:f(x)+i, c, c)
if b-a > d-b:
yield b
a, b, c=b, c, d
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CROSSREFS
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KEYWORD
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nonn,changed
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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