A030052 -id:A030052 - cp's OEIS frontend

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

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

A350430 a(n) is the smallest n-th power which can be represented as the sum of n distinct positive n-th powers in exactly n ways, or -1 if none exists.

Original entry on oeis.org

1, 625, 157464

Offset: 1

Ilya Gutkovskiy, Dec 30 2021

From Jon E. Schoenfield, Dec 30 2021: (Start)
222000^4 < a(4) < 4891341^4 = lcm(2829, 12259, 16359, 30381)^4 (see A039664, including the Wroblewski link).
10000^5 <= a(5) < 12528^5 = lcm(72, 1044, 1392, 2088, 3132)^5 (see A063923, including the Waldby link; note that, although the terms of A063923 include 72, 144, 1044, 1392, and 2088, whose LCM is only 4176, the primitive solution in which the sum of 5 distinct 5th powers is 144^5 is 0^5 + 27^5 + 84^5 + 110^5 + 133^5 = 144^5, which is not the sum of 5 positive n-th powers).
Conjecture: a(6) = -1. (End)

			For n = 2: 625 = 25^2 = 7^2 + 24^2 = 15^2 + 20^2.
For n = 3: 157464 = 54^3 = 6^3 + 36^3 + 48^3 = 12^3 + 19^3 + 53^3 = 27^3 + 36^3 + 45^3.

Cf. A007666, A025303, A025401, A030052, A039664, A063923, A146756, A230718, A331675.

cp's OEIS Frontend

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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