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
Examples
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.
Links
- Wikipedia, Beal's conjecture
Programs
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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
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.
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