A336797 Numbers, not divisible by 3, whose squares have exactly 4 nonzero digits in base 3.

7, 14, 16, 17, 26, 35, 47, 68, 350, 3788

Offset: 1

Author

Michel Marcus, Jan 27 2021

Keywords

• nonn
• base
• more

Comments

Is this sequence infinite?

Next term, if it exists, is > 3^500. - James Rayman, Feb 05 2021

Examples

			7^2=49 in base 3 is 1211, so 7 is a term.
14^2=196 in base 3 is 21021, so 14 is a term.
		

Links

• Alessio Moscariello, On sparse perfect powers, arXiv:2101.10415 [math.NT], 2021. See Question 11 p. 9.

Crossrefs

Cf. A007089 (numbers in base 3), A160385.

Programs

Mathematica

Select[Range[4000], Mod[#, 3] > 0 && Length @ Select[IntegerDigits[#^2, 3], #1 > 0 &] == 4 &] (* Amiram Eldar, Jan 27 2021 *)

PARI

isok(n) = (n%3) && #select(x->x, digits(n^2, 3)) == 4;

Python

from gmpy2 import isqrt, is_square
import itertools
N = 1000
powers = [1]
a_list = []
while len(powers) < N: powers.append(3 * powers[-1])
def attempt(n):
    if is_square(n): a_list.append(int(isqrt(n)))
for A, B, C in itertools.combinations(powers[1:], 3):
    for a, b, c in itertools.product([1, 2], repeat=3):
            attempt(a*A + b*B + c*C + 1)
print(sorted(a_list)) # James Rayman, Feb 05 2021