A384439 -id:A384439 - 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]]

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

A385323 a(n) is the smallest prime p for which the Diophantine equation Sum_{i=1..n} (x_i)^3 = p^3 has a solution, where (x_i), i=1..n, is a strictly increasing sequence of positive integers, or -1 if no such prime exists.

Original entry on oeis.org

2, -1, 19, 13, 17, 13, 17, 17, 19, 19, 23, 23, 29, 29, 29, 31, 37, 37, 41, 41, 43, 47, 53, 53, 53, 59, 59, 59, 67, 67, 67, 71, 71, 79, 79, 79, 83, 89, 89, 97

Offset: 1

Jean-Marc Rebert, Jun 25 2025

The sequence increases monotonically for n>5. However, not all primes are present. Missing so far are the primes 3, 5, 7, 11, 61 and 73. - Robert G. Wilson v, Jul 20 2025

			a(3) = 19, since 19 is prime, 3 < 10 < 18 and 3^3 + 10^3 + 18^3 = 6859 = 19^3, and no smaller prime satisfies this property.
a(4) = 13, since 13 is prime, 3 < 5 < 7 < 12 and 3^3 + 5^3 + 7^3 + 12^3 = 2197 = 13^3, and no smaller prime satisfies this property.

Cf. A030052, A377372, A377739, A384439.

from itertools import combinations
from sympy import nextprime
def A385323(n):
    if n == 2: return -1
    p = 2
    while True:
        p3 = p**3
        for k in combinations(range(1,p+1),n):
            if sum(i**3 for i in k) == p3:
                return p
        p = nextprime(p) # Chai Wah Wu, Jul 06 2025

a(21)-a(33) from Sean A. Irvine, Jul 05 2025

a(34)-a(40) from Robert G. Wilson v, Jul 20 2025

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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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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A385323 a(n) is the smallest prime p for which the Diophantine equation Sum_{i=1..n} (x_i)^3 = p^3 has a solution, where (x_i), i=1..n, is a strictly increasing sequence of positive integers, or -1 if no such prime exists.

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