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 A293694 #20 May 25 2025 09:23:15 %S A293694 20,34,65,135,320,369,544,1040,1095,1305,1350,1404,1620,1625,1746, %T A293694 1971,2056,2160,2379,2754,3060,3281,3996,4100,4470,5120,5265,5904, %U A293694 6625,7825,7830,8194,8575,8704,8796,10250,10935,11125,11700,12500,13154,14500,15579 %N A293694 Numbers z such that x^2 + y^8 = z^2 for positive integers x and y. %C A293694 Let i, j and k be nonnegative integers, m > n be positive integers. As ((m^2 - n^2)^(4*i+1) * (2*m*n)^(4*j+3) * (m^2 + n^2)^(4*k))^2 + ((m^2 - n^2)^i * (2*m*n)^(j+1) * (m^2 + n^2)^k)^8 = ((m^2 - n^2)^(4*i) * (2*m*n)^(4*j+3) * (m^2 + n^2)^(4*k+1))^2, so that the number of the form (m^2 - n^2)^(4*i) * (2*m*n)^(4*j+3) * (m^2 + n^2)^(4*k+1) is a term. %C A293694 When (x, y, z) is a solution of x^2 + y^4 = z^2 (i.e., z = A271576(n)), (x^(4*i+1) * y^(4*j+2) * z^(4*k), x^i * y^(j+1) * z^k, x^(4*i) * y^(4*j+2) * z^(4*k+1)) is a solution of x^2 + y^8 = z^2. %C A293694 When (x, y, z) is a solution of x^2 + y^6 = z^2 (i.e., z = A293690(n)), (x^(4*i+1) * y^(4*j+1) * z^(4*k), x^i * y^(j+1) * z^k, x^(4*i) * y^(4*j+1) * z^(4*k+1)) is a solution of x^2 + y^8 = z^2. %C A293694 When (x, y, z) is a solution of x^2 + y^8 = z^2, (x^(4*i+1) * y^(4*j) * z^(4*k), x^i * y^(j+1) * z^k, x^(4*i) * y^(4*j) * z^(4*k+1)) is also a solution of x^2 + y^8 = z^2. %H A293694 Karl-Heinz Hofmann, <a href="/A293694/b293694.txt">Table of n, a(n) for n = 1..13695</a> %e A293694 12^2 + 2^8 = 20^2, 20 is a term. %e A293694 63^2 + 2^8 = 65^2, 65 is a term. %t A293694 z[n_] := Count[n^2 - Range[(n^2 - 1)^(1/8)]^8, _?(IntegerQ[Sqrt[#]] &)] > 0; Select[Range[16000], z] %Y A293694 Cf. A009003, A228946, A271576, A293283, A293690, A293692. %K A293694 nonn %O A293694 1,1 %A A293694 _XU Pingya_, Oct 16 2017