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%I #20 Jan 17 2021 06:20:19
%S 4,9,25,225,676,2116,6724,7921,8100,16641,104329,131044,160801,176400,
%T 372100,389376,705600,4096576,7306209,7884864,47444544,146385801,
%U 254817369,373262400,607622500,895804900,1121580100,1330936324,1536875209,2097182025,2258435529,2749953600
%N Perfect powers such that the two immediately adjacent perfect powers both have an exponent greater than 2.
%C Apparently, all known terms (checked through 10^18) are squares with maximum exponent 2, i.e., terms of A111245 (squares that are not a higher power). This would imply that of 3 immediately adjacent perfect powers, at least one is a term of A111245. Is there a known counterexample of 3 consecutive perfect powers, none of which is in A111245?
%H Hugo Pfoertner, <a href="/A340642/b340642.txt">Table of n, a(n) for n = 1..181</a> (terms < 10^18)
%e The first terms, assuming 1 being at least a cube:
%e .
%e n p1 x^p1 p2 a(n) p3 z^p3
%e =y^p2
%e 1 >2 1 2 4 3 8
%e 2 3 8 2 9 4 16
%e 3 4 16 2 25 3 27
%e 4 3 216 2 225 5 243
%e 5 4 625 2 676 6 729
%o (PARI) a340642(limit)={my(p2=999, p1=2, n2=1, n1=4); for(n=5, limit, my(p0=ispower(n)); if(p0>1, if(p2>2&p0>2, print1(n1,", ")); n2=n1; n1=n; p2=p1; p1=p0))};
%o a340642(50000000)
%Y Cf. A001597, A025479, A076467, A097054, A111245, A153158, A340643, A340700, A340701.
%K nonn
%O 1,1
%A _Hugo Pfoertner_, Jan 14 2021
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