A346497 -id:A346497 - cp's OEIS frontend

A004642 Powers of 2 written in base 3.

1, 2, 11, 22, 121, 1012, 2101, 11202, 100111, 200222, 1101221, 2210212, 12121201, 102020102, 211110211, 1122221122, 10022220021, 20122210112, 111022121001, 222122012002, 1222021101011, 10221112202022, 21220002111121, 120210012000012, 1011120101000101, 2100010202000202

Offset: 0

N. J. A. Sloane

When n is odd, a(n) ends in 1, and when n is even, a(n) ends in 2, since 2^n is congruent to 1 mod 3 when n is odd and to 2 mod 3 when n is even. - Alonso del Arte Dec 11 2009
Sloane (1973) conjectured a(n) always has a 0 between the most and least significant digits if n > 15 (see A102483 and A346497).
Erdős (1978) conjectured that for n > 8 a(n) has at least one 2 (see link to Terry Tao's blog). - Dmitry Kamenetsky, Jan 10 2017

N. J. A. Sloane, The Persistence of a Number, J. Recr. Math. 6 (1973), 97-98.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Yagub N. Aliyev, Digits of powers of 2 in ternary numeral system, Notes on Number Theory and Discrete Mathematics, Vol. 29, No. 3 (2023), 474-485.
Paul Erdős, Some unconventional problems in number theory, Mathematics Magazine, Vol. 52, No. 2 (1979), pp. 67-70.
Donald L. Kreher and Douglas R. Stinson, On min-base palindromic representations of powers of 2, arXiv:2401.07351 [math.NT], 2024. See Table 4 p. 10.
Jeffrey C. Lagarias, Ternary Expansions of Powers of 2, Journal of the London Mathematical Society, Vol. 79, No. 3 (2009), pp. 562-588; arXiv preprint, arXiv:math/0512006 [math.DS], 2005-2008.
Terry Tao, The Collatz Conjecture, Littlewood-Offord theory, and powers of 2 and 3, 2011.
Eric Weisstein's World of Mathematics, Ternary.

Cf. A000079: powers of 2 written in base 10.
Cf. A004643, ..., A004655: powers of 2 written in base 4, 5, ..., 16.
Cf. A004656, A004658, A004659, ..., A004663: powers of 3 written in base 2, 4, 5, ..., 9.

[Seqint(Intseq(2^n, 3)): n in [0..30]]; // G. C. Greubel, Sep 10 2018

Table[FromDigits[IntegerDigits[2^n, 3]], {n, 25}] (* Alonso del Arte Dec 11 2009 *)

a(n)=fromdigits(digits(2^n,3)) \\ M. F. Hasler, Jun 23 2018

A102483 Numbers k such that 2^k contains no zeros in base 3.

Original entry on oeis.org

0, 1, 2, 3, 4, 15

Offset: 1

N. J. A. Sloane, Feb 25 2005

I conjectured in 1973 that there are no further terms. This question is still open.
A104320(a(n)) = 0. - Reinhard Zumkeller, Mar 01 2005
No other terms less than 200000. - Robert G. Wilson v, Dec 06 2005
a(7) > 10^7. - Martin Ehrenstein, Jul 27 2021
If it exists, a(7) > 10^21. - Robert Saye, Mar 23 2022

Robert I. Saye, On two conjectures concerning the ternary digits of powers of two, J. Integer Seq. 25 (2022) Article 22.3.4.
N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.
Eric Weisstein's World of Mathematics, Ternary

Cf. A007377, A004642, A346497.

Select[ Range@1000, FreeQ[ IntegerDigits[2^#, 3], 0] &] (* Robert G. Wilson v, Dec 06 2005 *)

for (n=0, 100, if (vecmin(digits(2^n, 3)), print1(n, ", "))) \\ Michel Marcus, Mar 25 2015

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

Original entry on oeis.org

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

Robert Saye, Feb 25 2022

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).

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

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

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}]

a(n) = my(k=1); while(!vecmin(Vec(Vecrev(digits(2^k,3)), n)), k++); k; \\ Michel Marcus, Feb 26 2022

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

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

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A004642 Powers of 2 written in base 3.

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A102483 Numbers k such that 2^k contains no zeros in base 3.

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

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Mathematica

PARI

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