A293694 -id:A293694 - cp's OEIS frontend

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

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]

A293695 Numbers z such that x^2 + y^8 = z^2 (with positive integers x and y) and gcd(x, y, z) = 1.

Original entry on oeis.org

65, 3281, 6625, 16385, 22945, 195313, 198593, 390689, 407009, 419905, 810529, 2882401, 2885681, 3077713, 4194305, 4200865, 4584929, 5764865, 5781185, 6184705, 9959105, 21523361, 21718673, 24405761, 25000001, 25006561, 30764801, 43046785, 43063105, 47241025

Offset: 1

XU Pingya, Oct 16 2017

nonn

Subsequence of A293694.

			63^2 + 2^8 = 65^2 and gcd(63, 2, 65) = 1, 65 is a term.
24999999^2 + 10^8 = 25000001^2 and gcd(24999999, 10, 25000001) = 1, 25000001 is a term.

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

Cf. A293284, A293691, A293693, A293694.

z={}; Do[If[IntegerQ[(n^2 - y^8)^(1/2)] && GCD[y, n]==1, AppendTo[z, n]], {n, 5.1*10^7}, {y, (n^2 - 1)^(1/8)}]; z

cp's OEIS Frontend

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

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