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.
Original entry on oeis.org
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
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.
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Table[SelectFirst[Table[{Length@Select[PowersRepresentations[p^3,4,3],#[[4]]>#[[3]]>#[[2]]>#[[1]]>0&],p},{p,Prime@Range@25}],#[[1]]==k&],{k,5}]
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
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.
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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
Showing 1-2 of 2 results.
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