A293690 Numbers z such that x^2 + y^6 = z^2 for positive integers x and y.

10, 17, 45, 80, 123, 136, 225, 234, 260, 270, 291, 325, 360, 365, 459, 510, 514, 640, 666, 745, 984, 1025, 1088, 1215, 1225, 1250, 1305, 1450, 1466, 1565, 1740, 1753, 1800, 1872, 1950, 1970, 2022, 2080, 2125, 2160, 2328, 2600, 2628, 2880, 2920, 3172, 3185

Offset

1,1

Comments

Let i, j and k are nonegtive integers, m > n are positive integers. As [(m^2 - n^2)^(3i+1) * (2mn)^(3j + 2) * (m^2 + n^2)^(3k)]^2 + [(m^2 - n^2)^i * (2mn)^(j + 1) * (m^2 + n^2)^k]^6 = [(m^2 - n^2)^(3i) * (2mn)^(3j + 2) * (m^2 + n^2)^(3k+1)]^2, so that number of form (m^2 - n^2)^(3i) * (2mn)^(3j + 2) * (m^2 + n^2)^(3k+1) is a term in sequence.

When (x, y, z) is solution of x^2 + y^4 = z^2 (i.e., z = A271576(n)), (x^(3i+1) * y^(3j + 1) * z^(3k), (x^i * y^(j + 1) * z^k)^6, x^(3i) * y^(3j + 1) * z^(3k+1) is solution of x^2 + y^6 = z^2.

When (x, y, z) is solution of x^2 + y^6 = z^2, (x^(3i+1) * y^(3j) * z^(3k), x^(3i) * y^(j + 1) * z^k, x^(3i) * y^(3j) * z^(3k+1)) is also.

Links

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

Example

6^2 + 2^6 = 10^2, 10 is a term.
15^2 + 2^6 = 17^2, 17 is a term.

Mathematica

z[n_] := Count[n^2 - Range[(n^2 - 1)^(1/6)]^6, _?(IntegerQ[Sqrt[#]] &)] > 0; Select[Range[3200], z]

Crossrefs

Cf. A009003, A070745, A228946, A271576, A274035.

Sequence in context: A019991 A164285 A268183 * A069546 A065018 A039336

Adjacent sequences: A293687 A293688 A293689 * A293691 A293692 A293693

Keyword

nonn

Author

XU Pingya, Oct 14 2017

Status

approved