A351928 Smallest positive integer k such that 2^k has no '2' in the last n digits of its ternary expansion.

2, 2, 6, 8, 8, 8, 20, 24, 24, 24, 72, 186, 186, 332, 332, 1134, 1134, 1134, 1134, 1134, 1134, 25458, 25458, 25458, 25458, 25458, 25458, 159140, 249968, 249968, 249968, 249968, 249968, 249968, 249968, 249968, 9076914, 9076914, 9076914, 9076914, 9076914, 9076914, 90062678

Offset

1,1

Comments

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

Erdős (~1978) conjectured that 1, 4, and 256 are the only powers of two whose ternary expansion consists solely of 0's and 1's.

Links

Table of n, a(n) for n=1..43.

Paul Erdős, Some unconventional problems in number theory, Mathematics Magazine, Vol. 52, No. 2 (1979), pp. 67-70.

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], 2], ++k]; k]; Table[smallest[n], {n, 1, 20}]

Prog

(PARI) a(n) = my(k=max(1, logint(3^(n-1), 2))); while(#select(x->(x==2), 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 2 in digits(pow2, 3)[-n:]:
        k += 1
        pow2 *= 2
    return k
an = 0
for n in range(1, 22):
    an = a(n, an)
    print(an, end=", ") # Michael S. Branicky, Feb 27 2022
(Python)
from itertools import count
def A351928(n):
    kmax, m = 3**n, (3**(n-1)).bit_length()
    k2 = pow(2, m, kmax)
    for k in count(m):
        a = k2
        while a > 0:
            a, b = divmod(a, 3)
            if b == 2:
                break
        else:
            return k
        k2 = 2*k2 % kmax # Chai Wah Wu, Mar 19 2022

Crossrefs

Cf. A004642, A117971, A351927.

Sequence in context: A334518 A099490 A167878 * A331786 A320139 A033724

Adjacent sequences: A351925 A351926 A351927 * A351929 A351930 A351931

Keyword

nonn,base

Author

Robert Saye, Feb 25 2022

Status

approved