A293692 Numbers z such that x^2 + y^7 = z^2 for positive integers x and y.

12, 18, 33, 54, 126, 160, 272, 366, 375, 520, 531, 540, 594, 630, 756, 825, 945, 1028, 1044, 1094, 1350, 1372, 1506, 1536, 1575, 1980, 2050, 2219, 2304, 2619, 2940, 3250, 3500, 3645, 3906, 3925, 4097, 4224, 4390, 4625, 5500, 5844, 5988, 6048, 6192, 6283, 6422

Offset

1,1

Comments

Let i, j and k are nonegtive 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.

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.

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.

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.

Links

Table of n, a(n) for n=1..47.

Example

4^2 + 2^7 = 12^2, 12 is a term.
31^2 + 2^7 = 33^2, 33 is a term.

Mathematica

z[n_] := Count[n^2 - Range[(n^2 - 1)^(1/7)]^7, _?(IntegerQ[Sqrt[#]] &)] > 0; Select[Range[6550], z]

Crossrefs

Cf. A070745, A228946, A293284.

Sequence in context: A277725 A319746 A079479 * A349027 A348965 A162694

Adjacent sequences: A293689 A293690 A293691 * A293693 A293694 A293695

Keyword

nonn

Author

XU Pingya, Oct 14 2017

Status

approved