A293692 -id:A293692 - 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]

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

Original entry on oeis.org

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

XU Pingya, Oct 14 2017

nonn

Let i, j and k be nonnegative integers, m > n be positive integers. As ((m^2 - n^2)^(3*i+1) * (2*m*n)^(3*j+2) * (m^2 + n^2)^(3*k))^2 + ((m^2 - n^2)^i * (2*m*n)^(j+1) * (m^2 + n^2)^k)^6 = ((m^2 - n^2)^(3*i) * (2*m*n)^(3*j+2) * (m^2 + n^2)^(3*k+1))^2, so that the number of the form (m^2 - n^2)^(3*i) * (2*m*n)^(3*j+2) * (m^2 + n^2)^(3*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^(3*i+1) * y^(3*j+1) * z^(3*k), x^i * y^(j+1) * z^k, x^(3*i) * y^(3*j+1) * z^(3*k+1)) is a solution of x^2 + y^6 = z^2.
When (x, y, z) is a solution of x^2 + y^6 = z^2, (x^(3*i+1) * y^(3*j) * z^(3*k), x^i * y^(j+1) * z^k, x^(3*i) * y^(3*j) * z^(3*k+1)) is also a solution of x^2 + y^6 = z^2.

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

Cf. A009003, A070745, A228946, A271576, A274035, A293692, A293694.

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

A293693 Numbers z such that x^2 + y^7 = z^2 (with positive integers x and y) and gcd(x, y, z) = 1.

Original entry on oeis.org

33, 1094, 2219, 4097, 6283, 39063, 40156, 69985, 78157, 82221, 148109, 411772, 412865, 450834, 524289, 526475, 602413, 823575, 827639, 893527, 1347831, 2391485, 2430547, 2500001, 2502187, 2803256, 3323543, 4783001, 4787065, 5307257, 7282969, 8957953, 9036077

Offset: 1

XU Pingya, Oct 14 2017

nonn

Subsequence of A293692.

			31^2 + 2^7 = 33^2 and gcd(31, 2, 33) = 1, 33 is a term.
8879827^2 + 60^7 = 9036077^2 and gcd(8879827, 60, 9036077) = 1, 9036077 is a term.

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

Cf. A103156, A174115, A293284, A293692.

z={};Do[If[IntegerQ[(n^2 - y^7)^(1/2)] && GCD[y,n]==1,AppendTo[z,n]],{n,9.7*10^6},{y,(n^2 - 1)^(1/7)}];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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A293690 Numbers z such that x^2 + y^6 = z^2 for positive integers x and y.

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A293693 Numbers z such that x^2 + y^7 = z^2 (with positive integers x and y) and gcd(x, y, z) = 1.

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Mathematica