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 A293690 #12 May 24 2025 06:26:57 %S A293690 10,17,45,80,123,136,225,234,260,270,291,325,360,365,459,510,514,640, %T A293690 666,745,984,1025,1088,1215,1225,1250,1305,1450,1466,1565,1740,1753, %U A293690 1800,1872,1950,1970,2022,2080,2125,2160,2328,2600,2628,2880,2920,3172,3185 %N A293690 Numbers z such that x^2 + y^6 = z^2 for positive integers x and y. %C A293690 Let i, j and k be nonnegative integers, m > n be positive integers. As ((m^2 - n^2)^(3*i+1) * (2*m*n)^(3*j+2) * (m^2 + n^2)^(3*k))^2 + ((m^2 - n^2)^i * (2*m*n)^(j+1) * (m^2 + n^2)^k)^6 = ((m^2 - n^2)^(3*i) * (2*m*n)^(3*j+2) * (m^2 + n^2)^(3*k+1))^2, so that the number of the form (m^2 - n^2)^(3*i) * (2*m*n)^(3*j+2) * (m^2 + n^2)^(3*k+1) is a term. %C A293690 When (x, y, z) is a solution of x^2 + y^4 = z^2 (i.e., z = A271576(n)), (x^(3*i+1) * y^(3*j+1) * z^(3*k), x^i * y^(j+1) * z^k, x^(3*i) * y^(3*j+1) * z^(3*k+1)) is a solution of x^2 + y^6 = z^2. %C A293690 When (x, y, z) is a solution of x^2 + y^6 = z^2, (x^(3*i+1) * y^(3*j) * z^(3*k), x^i * y^(j+1) * z^k, x^(3*i) * y^(3*j) * z^(3*k+1)) is also a solution of x^2 + y^6 = z^2. %e A293690 6^2 + 2^6 = 10^2, 10 is a term. %e A293690 15^2 + 2^6 = 17^2, 17 is a term. %t A293690 z[n_] := Count[n^2 - Range[(n^2 - 1)^(1/6)]^6, _?(IntegerQ[Sqrt[#]] &)] > 0; Select[Range[3200], z] %Y A293690 Cf. A009003, A070745, A228946, A271576, A274035, A293692, A293694. %K A293690 nonn %O A293690 1,1 %A A293690 _XU Pingya_, Oct 14 2017