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

Author

N. J. A. Sloane

Keywords

• nonn
• base
• easy

Comments

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

References

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

Links

• 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.

Crossrefs

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.

Programs

Magma

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

Mathematica

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

PARI

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