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 A178612 #18 Aug 27 2025 01:01:36
%S A178612 5,20,32,41,124,133,140,160,189,224,257,265,284,292,305,320,445,509,
%T A178612 581,644,673,945,985,1076,1085,1120,1280,1345,1436,1489,1541,1597,
%U A178612 1708,1772,1917,2048,2237,2273,2336,2345,2489,2624,2749,2889,2980,3105,3140,3205
%N A178612 Positive numbers of the form p^6 - 4*p^4*q + 4*p^2*q^2 + 4*q^3 (and p*q <> 0).
%C A178612 Conjecture: There are no perfect squares in this sequence (in spite of all numbers being congruent to 0 or 1 mod 4).
%C A178612 If any perfect square occurred in this sequence then a septic trinomial x^7 + A*x^2 + B with two irreducible factors of degree 3 and 4 would exist.
%C A178612 This sequence is a subsequence of A079896.
%H A178612 Robin Visser, <a href="/A178612/b178612.txt">Table of n, a(n) for n = 1..10000</a>
%t A178612 aa = {}; Do[Do[kk = p^6 - 4 p^4 q + 4 p^2 q^2 + 4 q^3; If[(kk > 0) && (p q != 0), AppendTo[aa, kk]], {p, 1, 200}], {q, -200, 200}]; Take[Union[aa], 100]
%o A178612 (Magma)
%o A178612 is_A178612 := function(k)
%o A178612 R<x> := PolynomialRing(Integers());
%o A178612 for s in Solutions(Thue(x^3 - 4*x^2 + 4*x + 4), k) do
%o A178612 if (s[1]*s[2]) ne 0 and IsSquare(s[1]) then return true; end if;
%o A178612 end for;
%o A178612 return false;
%o A178612 end function;
%o A178612 [k : k in [1..1000] | is_A178612(k)]; // _Robin Visser_, Aug 26 2025
%Y A178612 Cf. A079896.
%K A178612 nonn,changed
%O A178612 1,1
%A A178612 _Artur Jasinski_, May 30 2010