A293694 - OEIS

%I #13 Nov 14 2021 01:15:00

%S 20,34,65,135,320,369,544,1040,1095,1305,1350,1404,1620,1625,1746,

%T 1971,2056,2160,2379,2754,3060,3281,3996,4100,4470,5120,5265,5904,

%U 6625,7825,7830,8194,8575,8704,8796,10250,10935,11125,11700,12500,13154,14500,15579

%N Numbers z such that x^2 + y^8 = z^2 for positive integers x and y.

%C Let i, j and k are nonegtive integers, m > n are positive integers. As [(m^2 - n^2)^(4i+1) * (2mn)^(4j + 2) * (m^2 + n^2)^(4k)]^2 + [(m^2 - n^2)^i * (2mn)^(j + 1) * (m^2 + n^2)^k]^8 = [(m^2 - n^2)^(4i) * (2mn)^(4j + 2) * (m^2 + n^2)^(4k+1)]^2, so that number of form (m^2 - n^2)^(4i) * (2mn)^(4j + 2) * (m^2 + n^2)^(4k+1) is a term in sequence.

%C When (x, y, z) is solution of x^2 + y^4 = z^2 (i.e., z = A271576(n)), (x^(4i+1) * y^(4j + 2) * z^(4k), x^i * y^(j + 1) * z^k, x^(4i) * y^(4j + 1) * z^(4k+1)) is solution of x^2 + y^8 = z^2.

%C When (x, y, z) is solution of x^2 + y^6 = z^2 (i.e., z = A293690(n)), (x^(4i+1) * y^(4j + 1) * z^(4k), x^i * y^(j + 1) * z^k, x^(4i) * y^(4j + 1) * z^(4k+1) is solution of x^2 + y^8 = z^2.

%C When (x, y, z) is solution of x^2 + y^8 = z^2, (x^(4i+1) * y^(4j) * z^(4k), x^(4i) * y^(j + 1) * z^k, x^(4i) * y^(4j) * z^(4k+1)) is also

%H Karl-Heinz Hofmann, <a href="/A293694/b293694.txt">Table of n, a(n) for n = 1..13695</a>

%e 12^2 + 2^8 = 20^2, 20 is a term.

%e 63^2 + 2^8 = 65^2, 65 is a term.

%t z[n_] := Count[n^2 - Range[(n^2 - 1)^(1/8)]^8, _?(IntegerQ[Sqrt[#]] &)] > 0; Select[Range[16000], z]

%Y Cf. A009003, A228946, A271576, A293283, A293690, A293692.

%K nonn

%O 1,1

%A _XU Pingya_, Oct 16 2017