A351927 Smallest positive integer k such that 2^k has no '0' in the last n digits of its ternary expansion.

1, 2, 4, 10, 15, 15, 15, 15, 15, 15, 50, 50, 101, 101, 101, 101, 143, 143, 143, 143, 143, 143, 143, 143, 143, 1916, 1916, 1916, 1916, 1916, 1916, 82286, 1134022, 1639828, 3483159, 3483159, 3483159, 3917963, 3917963, 3917963, 4729774, 4729774, 9827775, 9827775, 43622201, 43622201, 43622201

Offset

1,2

Comments

The powers of two are required to have at least n ternary digits, i.e., 2^k >= 3^(n-1).

Sloane (1973) conjectured that every power 2^n with n > 15 has a '0' somewhere in its ternary expansion (see A102483 and A346497).

Links

Table of n, a(n) for n=1..47.

Robert I. Saye, On two conjectures concerning the ternary digits of powers of two, arXiv:2202.13256 [math.NT], 2022.

Mathematica

smallest[n_] := Module[{k}, k = Max[1, Ceiling[(n - 1) Log[2, 3]]];  While[MemberQ[Take[IntegerDigits[2^k, 3], -n], 0], ++k]; k]; Table[smallest[n], {n, 1, 20}]

Prog

(PARI) a(n) = my(k=1); while(!vecmin(Vec(Vecrev(digits(2^k, 3)), n)), k++); k; \\ Michel Marcus, Feb 26 2022
(Python)
from sympy.ntheory.digits import digits
def a(n, startk=1):
    k = max(startk, len(bin(3**(n-1))[2:]))
    pow2 = 2**k
    while 0 in digits(pow2, 3)[-n:]:
        k += 1
        pow2 *= 2
    return k
an = 0
for n in range(1, 32):
    an = a(n, an)
    print(an, end=", ") # Michael S. Branicky, Mar 10 2022
(Python)
from itertools import count
def A351927(n):
    kmax, m = 3**n, (3**(n-1)).bit_length()
    k2 = pow(2, m, kmax)
    for k in count(m):
        a = k2
        if 3*a >= kmax:
            while a > 0:
                a, b = divmod(a, 3)
                if b == 0:
                    break
            else:
                return k
        k2 = 2*k2 % kmax # Chai Wah Wu, Mar 19 2022

Crossrefs

Cf. A004642, A117970, A102483, A346497, A351928.

Sequence in context: A056392 A080149 A217134 * A333619 A128513 A048200

Adjacent sequences: A351924 A351925 A351926 * A351928 A351929 A351930

Keyword

nonn,base

Author

Robert Saye, Feb 25 2022

Status

approved