OFFSET
1,2
COMMENTS
It is unknown whether a(n) exists for all n (at least one sum is a square), but it is expected. It is also unknown whether the sequence is strictly increasing.
a(n) exists for n <= 11. - Chai Wah Wu, May 13 2024
LINKS
Bryle Morga et al., MathOverflow post about this sequence
EXAMPLE
a(3) = 1089 = 33^2 = 1^3 + 4^5 + 2^6.
a(8) = 4187530604721424 = 64711132^2 = 14^1 + 16^2 + 5^3 + 4^6 + 7^8 + 9^12 + 10^13 + 11^15.
PROG
(Python)
import itertools
import math
def a(n):
res = 0
for p in itertools.permutations([i for i in range(1, 2*n + 1)], n):
i = 0
s = 0
for j in range(1, 2*n + 1):
if j not in p:
s += p[i]**j
i += 1
if math.isqrt(s)**2 == s:
res = max(res, s)
return res
(Python)
from itertools import combinations, permutations
from sympy.ntheory.primetest import is_square
def A372653(n):
a, m, f = set(range(1, 2*n+1)), 0, [[b**c for c in range(2*n+1)] for b in range(2*n+1)]
for b in combinations(a, n):
clist = sorted(a-set(b))
for d in permutations(range(n)):
k = sum(f[b[i]][clist[d[i]]] for i in range(n))
if k>m and is_square(k):
m = k
return m # Chai Wah Wu, May 13 2024
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Bryle Morga, May 08 2024
EXTENSIONS
a(9) from Chai Wah Wu, May 13 2024
STATUS
approved