A282578 Least k such that k^n is the sum of two distinct proper prime powers (A246547), or 0 if no such k exists.

12, 5, 5, 3, 62

Offset

1,1

Comments

Corresponding values of k^n are 12, 25, 125, 81, 916132832, ...

Links

Table of n, a(n) for n=1..5.

Wikipedia, Beal's conjecture

Formula

a(p) <= 2 * (2^p - 1) where p is in A000043 since (2^p - 1)^p + (2^p - 1)^(p + 1) = (2 * (2^p - 1))^p.

Example

a(1) = 12 because 12 = 2^2 + 2^3.
a(2) = 5 because 5^2 = 2^4 + 3^2.
a(3) = 5 because 5^3 = 2^2 + 11^2.
a(4) = 3 because 3^4 = 2^5 + 7^2.
a(5) = 62 because 62^5 = 31^5 + 31^6.
a(9) = 2 because 2^9 = 7^3 + 13^2.

Prog

(Python)
from sympy import nextprime, perfect_power
def ppupto(limit): # distinct proper prime powers <= limit
    p = 2; p2 = pk = p*p; pklist = []
    while p2 <= limit:
        while pk <= limit: pklist.append(pk); pk *= p
        p = nextprime(p); p2 = pk = p*p
    return sorted(pklist)
def sum_of_pp(n):
    pp = ppupto(n); ppset = set(pp)
    for p in pp:
        if p > n//2: break
        if n - p in ppset and n - p != p: return True
    return False
def a(n):
    k = 2
    while not sum_of_pp(k**n): k += 1
    return k
print([a(n) for n in range(1, 6)]) # Michael S. Branicky, Dec 05 2021

Crossrefs

Cf. A001597, A225102, A246547, A282550.

Sequence in context: A157782 A335949 A002679 * A205141 A122561 A110185

Adjacent sequences: A282575 A282576 A282577 * A282579 A282580 A282581

Keyword

nonn,more

Author

Altug Alkan, Feb 20 2017

Status

approved