A352329 Squares in A030299.

1, 13527684, 34857216, 65318724, 73256481, 81432576, 139854276, 152843769, 157326849, 215384976, 245893761, 254817369, 326597184, 361874529, 375468129, 382945761, 385297641, 412739856, 523814769, 529874361, 537219684, 549386721, 587432169, 589324176, 597362481, 615387249

Offset: 1

Author

Luca Onnis and Marco Ripà, Mar 12 2022

Keywords

• nonn
• base

Comments

Conjecture: there are infinitely many terms.

References

• John D. Dixon and Brian Mortimer, Permutation groups. Graduate Texts in Mathematics, 163. Springer-Verlag, New York, 1996. xii+346 pp. ISBN: 0-387-94599-7 MR1409812 (98m:20003).

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..3185

Crossrefs

Cf. A030299, A240763.

Programs

Python

from itertools import permutations
def pmap(s, m): return sum(s[i-1]*10**(m-i) for i in range(1, len(s)+1))
def agen():
  m = 1
  while True:
    for s in permutations(range(1, m+1)): yield pmap(s, m)
    m += 1
def aupton(terms):
  alst, g = [], agen()
  while len(alst) < terms: alst += [next(g)]
  return alst
def is_perfect_square(n):
        return round(n ** 0.5) ** 2 == n
print([x for x in aupton(5000000) if is_perfect_square(x)])

Python

from itertools import count, islice, permutations
from sympy import integer_nthroot
def A352329_gen(): # generator of terms
    for l in count(1):
        if (r := l*(l+1)//2 % 9) == 0 or r == 1 or r == 4 or r == 7:
            m = tuple(10**(l-i-1) for i in range(l))
            for p in permutations(range(1,l+1)):
                if integer_nthroot(n := sum(prod(k) for k in zip(m,p)),2)[1]:
                    yield n
A352329_list = list(islice(A352329_gen(),10)) # Chai Wah Wu, Mar 21-22 2022