A383877 -id:A383877 - cp's OEIS frontend

A384439 a(n) is the smallest prime p such that the Diophantine equation x^3 + y^3 + z^3 + w^3 = p^3, where 0 < x < y < z < w has exactly n positive integer solutions.

23, 13, 59, 79, 97, 139, 163, 223, 151, 283, 251, 257, 263, 277, 227, 463, 271, 373, 587, 457, 641, 461, 499, 389, 503, 683, 761, 673, 509, 523, 709, 631, 757, 619, 571, 691, 929, 727

Offset: 1

Zhining Yang, May 29 2025

a(39) > 4027 if it exists. - Chai Wah Wu, Jun 03 2025

			a(3)=59, because 59^3 = 13^3 + 21^3 + 41^3 + 50^3 = 14^3 + 19^3 + 44^3 + 48^3 = 21^3 + 23^3 + 26^3 + 55^3 and no prime less than 59 has 3 solutions.

Chai Wah Wu, All terms <= 4007.

Cf. A377372, A383877.

Table[SelectFirst[Table[{Length@Select[PowersRepresentations[p^3,4,3],#[[4]]>#[[3]]>#[[2]]>#[[1]]>0&],p},{p,Prime@Range@25}],#[[1]]==k&],{k,5}]

A384182 a(n) is the smallest integer k such that the Diophantine equation x^3 + y^3 + z^3 + w^3 = k^4, where 0 < x < y < z < w has exactly n integer solutions.

Original entry on oeis.org

6, 9, 15, 34, 20, 19, 66, 28, 36, 35, 26, 30, 355, 97, 44, 329, 151, 65, 590, 89, 48, 42, 129, 54, 70, 99, 56, 178, 580, 128, 110, 392, 107, 518, 63, 125, 90, 887, 242, 78, 100, 138, 105, 96, 235, 141, 281, 205, 326, 1094, 117, 108, 197, 860, 159, 174, 291, 134

Offset: 1

Zhining Yang, May 21 2025

nonn

a(131)>1600.

			a(3)=15, because 15^4 = 13^3 + 21^3 + 23^3 + 30^3 = 11^3 + 16^3 + 21^3 + 33^3 = 9^3 + 11^3 + 21^3 + 34^3 and no integer less than 15 has 3 solutions.

Zhining Yang, Table of n, a(n) for n = 1..130

Cf. A383877.

s=Table[{k, Length@Select[PowersRepresentations[k^4, 4, 3], 0<#[[1]]<#[[2]]<#[[3]]<#[[4]]&]}, {k, 50}];a=Table[SelectFirst[s, #[[2]]==k&], {k, 6}][[All, 1]]

A384430 a(n) is the smallest positive integer k such that the Diophantine equation x^3 + y^3 + z^3 + w^3 = k^5, where 0 < x < y < z < w has exactly n integer solutions.

Original entry on oeis.org

8, 9, 10, 13, 74, 23, 40, 88, 31, 22, 17, 56

Offset: 1

Zhining Yang, Jun 14 2025

a(13)>360.

			a(3)=10, because 10^5 = 6^3 + 24^3 + 34^3 + 36^3 = 12^3 + 16^3 + 34^3 + 38^3 = 10^3 + 20^3 + 30^3 + 40^3 and no integer less than 10 has 3 solutions.

Cf. A383877, A384182.

s=Table[{k, Length@Select[PowersRepresentations[k^5, 4, 3], 0<#[[1]]<#[[2]]<#[[3]]<#[[4]]&]}, {k, 20}]; a=Table[SelectFirst[s, #[[2]]==k&], {k, 4}][[All, 1]]

A384132 Integers k such that the Diophantine equation x^3 + y^3 + z^3 + w^3 = k^3, where 0 < x < y < z < w has no integer solutions.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 16, 17, 19, 21, 22, 25, 27, 29, 47, 58, 61, 71, 113, 121

Offset: 1

Zhining Yang, May 20 2025

Conjecture: a(27)=121 is the largest integer whose cube cannot be described as the sum of four distinct positive cubes.

			13 is not a term because 13^3 = 5^3 + 7^3 + 9^3 + 10^3 = 1^3 + 5^3 + 7^3 + 12^3.

Cf. A003327, A383877.

a=Select[Range@125,Length@Select[PowersRepresentations[#^3,4,3],0<#[[1]]<#[[2]]<#[[3]]<#[[4]]&]==0&]

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A384439 a(n) is the smallest prime p such that the Diophantine equation x^3 + y^3 + z^3 + w^3 = p^3, where 0 < x < y < z < w has exactly n positive integer solutions.

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A384182 a(n) is the smallest integer k such that the Diophantine equation x^3 + y^3 + z^3 + w^3 = k^4, where 0 < x < y < z < w has exactly n integer solutions.

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A384430 a(n) is the smallest positive integer k such that the Diophantine equation x^3 + y^3 + z^3 + w^3 = k^5, where 0 < x < y < z < w has exactly n integer solutions.

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A384132 Integers k such that the Diophantine equation x^3 + y^3 + z^3 + w^3 = k^3, where 0 < x < y < z < w has no integer solutions.

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