A293690 -id:A293690 - cp's OEIS frontend

A293694 Numbers z such that x^2 + y^8 = z^2 for positive integers x and y.

20, 34, 65, 135, 320, 369, 544, 1040, 1095, 1305, 1350, 1404, 1620, 1625, 1746, 1971, 2056, 2160, 2379, 2754, 3060, 3281, 3996, 4100, 4470, 5120, 5265, 5904, 6625, 7825, 7830, 8194, 8575, 8704, 8796, 10250, 10935, 11125, 11700, 12500, 13154, 14500, 15579

Offset: 1

XU Pingya, Oct 16 2017

nonn

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.
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.
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.
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.

			12^2 + 2^8 = 20^2, 20 is a term.
63^2 + 2^8 = 65^2, 65 is a term.

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..13695

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

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

A293691 Numbers z such that x^2 + y^6 = z^2 (with positive integers x and y) and gcd(x, y, z) = 1.

Original entry on oeis.org

17, 365, 745, 1025, 1753, 7813, 8177, 11665, 15641, 16649, 27289, 58825, 59189, 65537, 66265, 66637, 81161, 117665, 118673, 129313, 183185, 250001, 250729, 265721, 273533, 324545, 367649, 531457, 532465, 596977, 746497, 762121, 781441, 864145, 885781, 886145

Offset: 1

XU Pingya, Oct 14 2017

nonn

Subsequence of A293690.

			15^2 + 2^6 = 17^2 and gcd(15, 2, 17) = 1, 17 is a term.
885416^2 + 33^6 = 886145^2 and gcd(885416, 33, 886145) = 1, 886145 is a term.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A103156, A174115, A293690.

z={};Do[If[IntegerQ[(n^2 - y^6)^(1/2)] && GCD[y,n]==1,AppendTo[z,n]],{n,8.9*10^5},{y,(n^2 - 1)^(1/6)}];z

cp's OEIS Frontend

A293694 Numbers z such that x^2 + y^8 = z^2 for positive integers x and y.

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