A070067 -id:A070067 - cp's OEIS frontend

A070065 Values of x in positive integer solutions of x^2 + y^5 = z^3, listed in increasing order of z. (If a z-value occurs twice, list solutions in increasing order of y.)

10, 1242, 1024, 5632, 20480, 17920, 37376, 39151, 36599, 15552, 26001, 75000, 124416, 68608, 172800, 209952, 359424, 413343, 579096, 327680, 684288, 1090625, 1306368, 1650240, 1506463, 1529437, 1607445, 1525899, 3637224, 2783744

Offset: 1

Dean Hickerson and Dan Asimov (asimov(AT)msri.org), Apr 18 2002

nonn

If x is in the sequence, then so is c^15 x for positive integers c, since if (x,y,z) is a solution, so is (c^15 x, c^6 y, c^10 z). - Robert Israel, Jul 26 2017

			The first 5 solutions are (x,y,z) = (10,3,7), (1242,9,117), (1024,16,128), (5632,16,320) and (20480,32,768).

y-values are in A070066, z-values are in A070067.

For[z=1, True, z++, z3=z^3; For[y=1, (d=z3-y^5)>0, y++, If[IntegerQ[x=Sqrt[d]], Print[{x, y, z}]]]]

A293283 Numbers n such that n^2 = a^2 + b^5 for positive integers a b and n.

Original entry on oeis.org

6, 9, 18, 40, 42, 68, 75, 90, 99, 105, 122, 126, 130, 174, 192, 196, 225, 251, 257, 288, 315, 325, 330, 350, 405, 490, 499, 504, 516, 528, 546, 550, 576, 614, 651, 665, 684, 726, 735, 744, 849, 882, 900, 920, 936, 974, 1025, 1032, 1036, 1107, 1140, 1183, 1200

Offset: 1

XU Pingya, Oct 04 2017

nonn

For n > 0, k = (n + 1)(2n + 1)^2 is a term in this sequence, because k^2 = (n * (2n + 1)^2)^2 + (2n + 1)^5. Examples: 18, 75, 196, 405, 726, 1183.
When z^2 = x^2 + y^2 (i.e., z = A009003(n)), (z * y^4)^2 = (x * y^4)^2 + (y^2)^5. Thus z * y^4 is a term in this sequence. For example, 1200. More generally, for positive integer i, j and k, x^(5i - 5) * y^(5j - 1) * z^(5k - 5) is in this sequence.
When z^2 = x^2 + y^3 (i.e., z = A070745(n)), (z * y)^2 = (x * y)^2 + y^5. Thus z * y is in this sequence. E.g. 6, 18, 40, ... . More generally, for positive integer i, j and k, x^(5i - 5) * y^(5j - 4) * z^(5k - 4) is in this sequence.
When z^2 = x^2 + y^4 (i.e., z = A271576(n)), (z * y^3)^2 = (x * y^3)^2 + (y^2)^5. Thus z * y^3 is also in this sequence. E.g. 40, 405, 1107, ... . More generally, for positive integer i, j and k, x^(5i - 5) * y^(5j - 2) * z^(5k - 4) is in this sequence.

			6^2 = 2^2 + 2^5.
9^2 = 7^2 + 2^5.

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

Cf. A009003, A070067, A070745, A228946, A271576, A274035.

c[n_]: = Count[n^2 - Range[(n^2 - 1)^(1/5)]^5, _?(IntegerQ[Sqrt[#]] &)] > 0;
Select[Range[1200], c]

isok(n) = for (k=1, n-1, if (ispower(n^2-k^2, 5), return (1));); return (0); \\ Michel Marcus, Oct 06 2017

A070066 Values of y in positive integer solutions of x^2 + y^5 = z^3, listed in increasing order of z. (If a z-value occurs twice, list solutions in increasing order of y.)

Original entry on oeis.org

3, 9, 16, 16, 32, 48, 24, 6, 55, 72, 72, 100, 72, 112, 72, 108, 144, 162, 36, 192, 72, 100, 216, 72, 295, 343, 351, 359, 72, 368, 423, 343, 216, 300, 343, 648, 800, 783, 625, 833, 400, 450, 648, 972, 496, 576, 1024, 864, 675, 972, 1215, 242, 72, 500, 1176

Offset: 1

Dean Hickerson and Dan Asimov (asimov(AT)msri.org), Apr 18 2002

nonn

			The first 5 solutions are (x,y,z) = (10,3,7), (1242,9,117), (1024,16,128), (5632,16,320) and (20480,32,768).

x-values are in A070065, z-values are in A070067.

For[z=1, True, z++, z3=z^3; For[y=1, (d=z3-y^5)>0, y++, If[IntegerQ[x=Sqrt[d]], Print[{x, y, z}]]]]

cp's OEIS Frontend

A070065 Values of x in positive integer solutions of x^2 + y^5 = z^3, listed in increasing order of z. (If a z-value occurs twice, list solutions in increasing order of y.)

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A293283 Numbers n such that n^2 = a^2 + b^5 for positive integers a b and n.

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A070066 Values of y in positive integer solutions of x^2 + y^5 = z^3, listed in increasing order of z. (If a z-value occurs twice, list solutions in increasing order of y.)

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