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.
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
Examples
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.
Programs
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Python
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
Extensions
a(21)-a(33) from Sean A. Irvine, Jul 05 2025
a(34)-a(40) from Robert G. Wilson v, Jul 20 2025
Comments