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 A293692 #17 May 24 2025 10:34:46 %S A293692 12,18,33,54,126,160,272,366,375,520,531,540,594,630,756,825,945,1028, %T A293692 1044,1094,1350,1372,1506,1536,1575,1980,2050,2219,2304,2619,2940, %U A293692 3250,3500,3645,3906,3925,4097,4224,4390,4625,5500,5844,5988,6048,6192,6283,6422 %N A293692 Numbers z such that x^2 + y^7 = z^2 for positive integers x and y. %C A293692 Let i, j and k are nonnegative integers, n is positive integer. As [(n^)^(7i+1) * (2n+1)^(7j + 3) * (n + 1)^(7k)]^2 + [n)^(2i) * (2n + 1)^(2j + 1) * (n + 1)^(2k)]^7 = [n^(7i) * (2n + 1)^(7j + 3) * (n+1)^(7k+1)]^2, so that number of form n^(7i) * (2n + 1)^(7j + 3) * (n+1)^(7k+1) is a term in sequence. %C A293692 When (x, y, z) is solution of x^2 + y^3 = z^2 (i.e., z = A070745(n)), (x^(7i+1) * y^(7j + 2) * z^(7k)]^2, x^(2i) * y^(2j + 1) * z^(2k), x^(7i) * y^(7j + 2) * z^(7k+1) is solution of x^2 + y^7 = z^2. %C A293692 When (x, y, z) is solution of x^2 + y^5 = z^2, (i.e., z = A293284(n)), x^(7i+1) * y^(7j + 1) * z^(7k), x^(2i) * y^(2j + 1) * z^(2k), x^(7i) * y^(7j + 1) * z^(7k+1) is solution of x^2 + y^7 = z^2. %C A293692 When (x, y, z) is solution of x^2 + y^7 = z^2, (x^(7i+1) * y^(7j + 2) * z^(7k), x^(7i) * y^(j + 1) * z^(7k), x^(7i) * y^(7j +2) * z^(7k)) is also. %C A293692 If x^2 + y^7 = z^2 then y^7 = z^2 - x^2 = (z - x)(z + x) and so (z - x, z + x) is a divisor pair of z^7. - _David A. Corneth_, May 24 2025 %e A293692 4^2 + 2^7 = 12^2, 12 is a term. %e A293692 31^2 + 2^7 = 33^2, 33 is a term. %t A293692 z[n_] := Count[n^2 - Range[(n^2 - 1)^(1/7)]^7, _?(IntegerQ[Sqrt[#]] &)] > 0; Select[Range[6550], z] %Y A293692 Cf. A070745, A228946, A293284. %K A293692 nonn,uned %O A293692 1,1 %A A293692 _XU Pingya_, Oct 14 2017