This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A340586 #6 Jan 14 2021 15:02:50
%S A340586 8,16,169,216,343,400,441,512,625,729,841,900,1156,1444,1521,1600,
%T A340586 1728,1849,1936,2048,2401,2601,2744,2916,3125,3249,3375,3600,3721,
%U A340586 3844,4096,4356,4489,4624,4761,4913,5184,5329,5476,5625,5832,6084,6241,6561,6859,7056
%N A340586 Perfect powers such that the two immediately adjacent perfect powers both have a largest exponent A025479 equal to 2.
%e A340586 a(1) = 8 because its neighboring perfect powers 4 = 2^2 and 9 = 3^2 both have the largest exponent 2.
%e A340586 9 is not in the sequence because both exponents of the neighboring perfect powers 8 = 2^3 and 16 = 2^4 are > 2.
%e A340586 a(2) = 16: neighbors 9 = 3^2 and 25 = 5^2 satisfy the exponent condition.
%e A340586 Next excluded terms: 25 (16 = 2^4, 27 = 3^3), 27 (32 = 2^5), 32 (27 = 3^3), 36 (32 = 2^5), 49 (64 = 2^6), 64 (81 = 3^4), 81 (64 = 2^6), 100 (81 = 3^4), 121 (125 = 5^3), 125 (128 = 2^7), 128 (125 = 5^3), 144 (128 = 2^7).
%e A340586 a(3) = 169: neighbors 144 = 12^2 and 196 = 14^2 satisfy the exponent condition.
%o A340586 (PARI) a340586(limit)={my(p2=999,p1=2,n2=1,n1=4);for(n=5,limit,my(p0=ispower(n));if(p0>1,if(p2+p0==4,print1(n1,", "));n2=n1;n1=n;p2=p1;p1=p0))};
%o A340586 a340586(7500)
%Y A340586 Cf. A000290, A001597, A025479, A111245, A153158, A340587, A340640, A340641.
%K A340586 nonn
%O A340586 1,1
%A A340586 _Hugo Pfoertner_, Jan 14 2021