A372652 Smallest 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, 1225, 25921, 372100, 3530641, 55890576, 1682148196

Offset

1,2

Comments

It is unknown whether a(n) exists for all n, but it is expected.

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 question about this sequence.

Example

a(5) = 25921 = 161^2 = 8^1 + 9^3 + 6^5 + 4^7 + 2^10.
a(6) = 372100 = 610^2 = 11^1 + 10^3 + 8^4 + 12^5 + 7^6 + 2^9.

Prog

(Python)
import itertools
import math
def a(n):
  res = math.inf
  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 = min(res, s)
  return res
(Python)
from itertools import combinations, permutations
from sympy.ntheory.primetest import is_square
def A372652(n):
    a, m, f = set(range(1, 2*n+1)), n*(2*n)**(2*n), [[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. A372653.

Sequence in context: A174636 A374143 A054344 * A048912 A036411 A372653

Adjacent sequences: A372649 A372650 A372651 * A372653 A372654 A372655

Keyword

nonn,more

Author

Bryle Morga, May 08 2024

Extensions

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

Status

approved