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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.

%I A385323 #29 Jul 30 2025 23:32:08
%S A385323 2,-1,19,13,17,13,17,17,19,19,23,23,29,29,29,31,37,37,41,41,43,47,53,
%T A385323 53,53,59,59,59,67,67,67,71,71,79,79,79,83,89,89,97
%N 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.
%C A385323 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
%e A385323 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.
%e A385323 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.
%o A385323 (Python)
%o A385323 from itertools import combinations
%o A385323 from sympy import nextprime
%o A385323 def A385323(n):
%o A385323     if n == 2: return -1
%o A385323     p = 2
%o A385323     while True:
%o A385323         p3 = p**3
%o A385323         for k in combinations(range(1,p+1),n):
%o A385323             if sum(i**3 for i in k) == p3:
%o A385323                 return p
%o A385323         p = nextprime(p) # _Chai Wah Wu_, Jul 06 2025
%Y A385323 Cf. A030052, A377372, A377739, A384439.
%K A385323 sign,more
%O A385323 1,1
%A A385323 _Jean-Marc Rebert_, Jun 25 2025
%E A385323 a(21)-a(33) from _Sean A. Irvine_, Jul 05 2025
%E A385323 a(34)-a(40) from _Robert G. Wilson v_, Jul 20 2025