cp's OEIS Frontend

A340642 Perfect powers such that the two immediately adjacent perfect powers both have an exponent greater than 2.

%I A340642 #20 Jan 17 2021 06:20:19
%S A340642 4,9,25,225,676,2116,6724,7921,8100,16641,104329,131044,160801,176400,
%T A340642 372100,389376,705600,4096576,7306209,7884864,47444544,146385801,
%U A340642 254817369,373262400,607622500,895804900,1121580100,1330936324,1536875209,2097182025,2258435529,2749953600
%N A340642 Perfect powers such that the two immediately adjacent perfect powers both have an exponent greater than 2.
%C A340642 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 A340642 Hugo Pfoertner, <a href="/A340642/b340642.txt">Table of n, a(n) for n = 1..181</a> (terms < 10^18)
%e A340642 The first terms, assuming 1 being at least a cube:
%e A340642 .
%e A340642   n   p1  x^p1  p2  a(n)  p3  z^p3
%e A340642                    =y^p2
%e A340642   1  >2     1   2     4   3     8
%e A340642   2   3     8   2     9   4    16
%e A340642   3   4    16   2    25   3    27
%e A340642   4   3   216   2   225   5   243
%e A340642   5   4   625   2   676   6   729
%o A340642 (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 a340642(50000000)
%Y A340642 Cf. A001597, A025479, A076467, A097054, A111245, A153158, A340643, A340700, A340701.
%K A340642 nonn
%O A340642 1,1
%A A340642 _Hugo Pfoertner_, Jan 14 2021