A228946 -id:A228946 - cp's OEIS frontend

A066648 Cubes of the form a^2 + b^3 with a, b > 0.

512, 1000, 2744, 21952, 32768, 35937, 64000, 175616, 185193, 274625, 357911, 373248, 405224, 474552, 729000, 1157625, 1404928, 1481544, 2000376, 2097152, 2197000, 2299968, 2744000, 3241792, 3652264, 3723875, 4096000, 5451776, 7189057, 8000000, 10218313, 10360232

Offset: 1

Reinhard Zumkeller, Dec 17 2001

nonn

			8^3 = a(0) = 512 = 169 + 343 = 13^2 + 7^3;
10^3 = a(1) = 1000 = 784 + 216 = 28^2 + 6^3.

Amiram Eldar, Table of n, a(n) for n = 1..5000

Cf. A055394, A066647, A228946.

q[n_] := Length[Reduce[a^2 + b^3 == n && a > 0 && b > 0, {a, b}, Integers]] > 0; Select[Range[220]^3, q] (* Amiram Eldar, Mar 20 2025 *)

a(n) = A228946(n)^3. - R. J. Mathar, Dec 03 2015

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

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

Original entry on oeis.org

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

XU Pingya, Oct 14 2017

Let i, j and k are nonnegative 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.
If x^2 + y^7 = z^2 then y^7 = z^2 - x^2 = (z - x)(z + x) and so (z - x, z + x) is a divisor pair of z^7. - David A. Corneth, May 24 2025

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

Cf. A070745, A228946, A293284.

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

A293694 Numbers z such that x^2 + y^8 = z^2 for positive integers x and y.

Original entry on oeis.org

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]

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A066648 Cubes of the form a^2 + b^3 with a, b > 0.

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

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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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