A377444 -id:A377444 - cp's OEIS frontend

A383877 a(n) is the smallest integer k such that the Diophantine equation x^3 + y^3 + z^3 + w^3 = k^3, where 0 < x < y < z < w has exactly n integer solutions, or 0 if there is no such k.

14, 13, 55, 26, 52, 63, 70, 66, 56, 104, 102, 143, 161, 91, 117, 112, 78, 236, 180, 217, 198, 192, 140, 292, 216, 259, 156, 196, 344, 168, 210, 264, 325, 252, 406, 360, 380, 402, 315, 338, 234, 308, 351, 182, 396, 408, 399, 432, 441, 312, 474, 636, 513, 273, 336, 476, 618, 666

Offset: 1

Zhining Yang, May 13 2025

nonn

The largest term for k<=10000 is a(3569)=9828.

Conjecture: a(n) != 0 for all n.

			a(3)=55, because 55^3 = 7^3 + 24^3 + 38^3 + 46^3 = 7^3 + 12^3 + 34^3 + 50^3 = 17^3 + 19^3 + 28^3 + 51^3 and no integer less than 55 has 3 solutions.

Zhining Yang, Table of n, a(n) for n = 1..1280

Cf. A377444, A384439.

s=Table[{k,Length@Select[PowersRepresentations[k^3,4,3],0<#[[1]]<#[[2]]<#[[3]]<#[[4]]&]},{k,100}];
a=Table[SelectFirst[s,#[[2]]==k&],{k,9}][[All,1]]

A385354 a(n) is the smallest positive integer k such that the Diophantine equation x^3 + y^3 + z^3 = k^2, where 0 < x < y < z has exactly n integer solutions.

Original entry on oeis.org

6, 188, 768, 1728, 2640, 21120, 42336, 13824, 71280, 5832, 80352, 74088, 425088, 421875, 1058400, 110592, 287496, 46656

Offset: 1

Zhining Yang, Jun 26 2025

a(19) > 2000000, a(20) = 216000, a(22) = 884736.

			a(3)=768, because 768^2 = 54^3 + 59^3 + 61^3 = 40^3 + 62^3 + 66^3 = 24^3 + 40^3 + 80^3 and no integer less than 768 has 3 solutions.

Cf. A024975, A025419, A377444.

s = Table[{k, Length@Select[PowersRepresentations[k^2, 3, 3],
     0 < #[[1]] < #[[2]] < #[[3]] &]}, {k, 2000}];
a = Table[SelectFirst[s, #[[2]] == k &], {k, 4}][[All, 1]]

a(18) from Chai Wah Wu, Jul 05 2025

A385566 a(n) is the smallest positive integer k such that the Diophantine equation x^3 + y^3 + z^3 = k^6, where 0 < x < y < z has exactly n integer solutions.

Original entry on oeis.org

3, 6, 16, 12, 27, 63, 38, 24, 94, 18, 123, 42, 93, 75, 141, 48, 66, 36, 153, 60, 140, 96, 279, 114, 200, 138, 410, 174, 72, 126, 186, 168, 204, 150, 108, 426, 132, 220, 418, 246, 498, 736, 144, 120, 294, 306, 210, 666, 282, 378, 252, 770, 216, 460, 462, 534, 180

Offset: 1

Zhining Yang, Jul 03 2025

nonn

			a(3)=16, because 16^6 = 9^3 + 58^3 + 255^3 = 9^3 + 183^3 + 220^3 = 22^3 + 57^3 + 255^3  and no integer less than 16 has 3 solutions.

Chai Wah Wu, Table of n, a(n) for n = 1..104

Cf. A024975, A025419, A377444, A385354, A385565.

s = Table[{k, Length@Select[PowersRepresentations[k^6, 3, 3], 0 < #[[1]] < #[[2]] < #[[3]] &]}, {k, 30}];
a = Table[SelectFirst[s, #[[2]] == k &], {k, 5}][[All, 1]]

a(41)-a(57) from Chai Wah Wu, Jul 07 2025

A377372 a(n) is the smallest prime p such that the Diophantine equation x^3 + y^3 + z^3 = p^3, where 0 < x <= y <= z has exactly n positive integer solutions.

Original entry on oeis.org

2, 19, 41, 479, 1031, 1181, 577, 2999, 10711, 29033, 24919, 49069, 60919, 169019, 209563, 254993, 337537

Offset: 0

Zhining Yang, Dec 28 2024

			a(3)=479, because 479^3 = 47^3 + 350^3 + 406^3 = 109^3 + 293^3 + 437^3 = 256^3 + 311^3 + 398^3 and no prime less than 479 has 3 solutions.

Cf. A316359, A377444, A384439.

a = Table[SelectFirst[Table[{p,Length@Select[PowersRepresentations[p^3, 3, 3], #[[1]] > 0 &]}, {p, Prime@Range@200}], #[[2]] == k &], {k, 0, 6}]

a(11) from Jinyuan Wang, May 31 2025
a(12) from Chai Wah Wu, Jun 03 2025
a(13)-a(15) from Chai Wah Wu, Jun 04 2025
a(16) from Chai Wah Wu, Jun 10 2025

A385565 a(n) is the smallest positive integer k such that the Diophantine equation x^3 + y^3 + z^3 = k^4, where 0 < x < y < z has exactly n integer solutions.

Original entry on oeis.org

11, 21, 64, 144, 330, 846, 342, 252, 1331, 1008, 720, 1890, 3780, 729, 4200, 2016, 1000, 216, 6300, 8352, 10800, 12312, 8568, 19440, 8280, 9576, 21204

Offset: 1

Zhining Yang, Jul 03 2025

a(13) and a(15) not found up to k = 3300, a(14) = 729, a(16) = 2016, a(17) = 1000, a(18) = 216.

			a(3)=64, because 64^4 = 9^3 + 58^3 + 255^3 = 9^3 + 183^3 + 220^3 = 22^3 + 57^3 + 255^3 and no integer less than 64 has 3 solutions.

Cf. A024975, A025419, A377444, A385354, A385566.

s = Table[{k, Length@Select[PowersRepresentations[k^4, 3, 3],
      0 < #[[1]] < #[[2]] < #[[3]] &]}, {k, 400}];
a = Table[SelectFirst[s, #[[2]] == k &], {k, 5}][[All, 1]]

a(13), a(15), a(19)-a(21) from Chai Wah Wu, Jul 08 2025

a(22)-a(27) from Chai Wah Wu, Jul 18 2025

A385409 a(n) is the smallest positive integer k such that the Diophantine equation x^3 + y^3 + z^3 + w^3 = k^2, where 0 < x < y < z < w has exactly n integer solutions.

Original entry on oeis.org

10, 42, 39, 153, 126, 276, 273, 312, 315, 476, 588, 336, 546, 777, 1053, 756, 1216, 1386, 1560, 1134, 1323, 1488, 1365, 1368, 1344, 1596, 2366, 2496, 2988, 1680, 2548, 1736, 2184, 3003, 3720, 2520, 3185, 3552, 2268, 3564, 4095, 3213, 4578, 4392, 5208, 4004, 4599, 5733

Offset: 1

Zhining Yang, Jun 27 2025

nonn

Conjecture: a(n) exists for all n.

			a(4)=153, because 153^2 = 5^3 + 15^3 + 21^3 + 22^3 = 2^3 + 7^3 + 15^3 + 27^3 = 6^3 + 8^3 + 9^3 + 28^3 = 1^3 + 5^3 + 11^3 + 28^3 and no integer less than 153 has 4 solutions.

Zhining Yang, Table of n, a(n) for n = 1..1341

Cf. A024975, A025419, A377444, A384430, A385354.

s = Table[{k, Length@Select[PowersRepresentations[k^2, 4, 3],
      0 < #[[1]] < #[[2]] < #[[3]] < #[[4]] &]}, {k, 500}];
a = Table[SelectFirst[s, #[[2]] == k &], {k, 10}][[All, 1]]

cp's OEIS Frontend

A383877 a(n) is the smallest integer k such that the Diophantine equation x^3 + y^3 + z^3 + w^3 = k^3, where 0 < x < y < z < w has exactly n integer solutions, or 0 if there is no such k.

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A385354 a(n) is the smallest positive integer k such that the Diophantine equation x^3 + y^3 + z^3 = k^2, where 0 < x < y < z has exactly n integer solutions.

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A385566 a(n) is the smallest positive integer k such that the Diophantine equation x^3 + y^3 + z^3 = k^6, where 0 < x < y < z has exactly n integer solutions.

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A377372 a(n) is the smallest prime p such that the Diophantine equation x^3 + y^3 + z^3 = p^3, where 0 < x <= y <= z has exactly n positive integer solutions.

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A385565 a(n) is the smallest positive integer k such that the Diophantine equation x^3 + y^3 + z^3 = k^4, where 0 < x < y < z has exactly n integer solutions.

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A385409 a(n) is the smallest positive integer k such that the Diophantine equation x^3 + y^3 + z^3 + w^3 = k^2, where 0 < x < y < z < w has exactly n integer solutions.

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Comments

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Mathematica