A372653 Largest square that can be expressed as Sum_{i=1..n} p(2i-1)^p(2i) where p is a permutation of {1, 2, 3, ..., 2n}.

1, 9, 1089, 73441, 100280196, 1977847729, 4146497327025, 4187530604721424, 121439588246116522561

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

Table of n, a(n) for n=1..9.

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

Cf. A372652.

Sequence in context: A372652 A048912 A036411 * A075412 A174253 A365595

Adjacent sequences: A372650 A372651 A372652 * A372654 A372655 A372656

Keyword

nonn,hard,more

Author

Bryle Morga, May 08 2024

Extensions

a(9) from Chai Wah Wu, May 13 2024

Status

approved