A004642 -id:A004642 - cp's OEIS frontend

A117970 Position of first 0 counting from the least significant digit in the ternary expansion of 2^n.

0, 0, 0, 0, 0, 3, 2, 2, 4, 4, 5, 4, 2, 2, 4, 0, 3, 4, 2, 2, 3, 3, 8, 3, 2, 2, 5, 5, 6, 5, 2, 2, 11, 4, 3, 5, 2, 2, 3, 3, 11, 3, 2, 2, 5, 8, 4, 5, 2, 2, 13, 5, 3, 5, 2, 2, 3, 3, 4, 3, 2, 2, 4, 4, 7, 4, 2, 2, 4, 6, 3, 4, 2, 2, 3, 3, 7, 3, 2, 2, 6, 6, 7, 6, 2, 2, 10, 4, 3, 7, 2, 2, 3, 3, 5, 3, 2, 2, 11, 5, 4, 17

Offset: 0

Eric W. Weisstein, Apr 06 2006

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Ternary

Cf. A004642, A102483, A104320, A117971.

a(n) = { my (p=2^n); for (k=1, oo, if (p==0, return (0), p%3==0, return (k), p\=3)) } \\ Rémy Sigrist, Dec 20 2019

Edited by Charles R Greathouse IV, Aug 05 2010

a(0) = 0 prepended by Rémy Sigrist, Dec 20 2019

A004645 Powers of 2 written in base 6.

Original entry on oeis.org

1, 2, 4, 12, 24, 52, 144, 332, 1104, 2212, 4424, 13252, 30544, 101532, 203504, 411412, 1223224, 2450452, 5341344, 15123132, 34250304, 112541012, 225522024, 455444052, 1355332144, 3155104332, 10354213104

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A000079, A004642, ..., A004655: powers of 2 written in base 10, 3, 4, ..., 16

Cf. A000244, A004656, A004658, A004659, ... : powers of 3 written in base 10, 2, 4, 5, ...

[Seqint(Intseq(2^n, 6)): n in [0..30]]; // G. C. Greubel, Sep 10 2018

Table[FromDigits[IntegerDigits[2^n, 6]], {n, 0, 40}] (* Vincenzo Librandi, Jun 07 2013 *)

a(n)=fromdigits(digits(2^n,6)) \\ M. F. Hasler, Jun 23 2018

A117971 One-based position of the first 2 from the least significant digit in the ternary expansion of 2^n, or 0 if there are no 2's present.

Original entry on oeis.org

0, 1, 0, 1, 2, 1, 4, 1, 0, 1, 2, 1, 3, 1, 3, 1, 2, 1, 5, 1, 8, 1, 2, 1, 11, 1, 11, 1, 2, 1, 3, 1, 3, 1, 2, 1, 4, 1, 4, 1, 2, 1, 5, 1, 4, 1, 2, 1, 3, 1, 3, 1, 2, 1, 6, 1, 8, 1, 2, 1, 4, 1, 7, 1, 2, 1, 3, 1, 3, 1, 2, 1, 12, 1, 7, 1, 2, 1, 6, 1, 10, 1, 2, 1, 3, 1, 3, 1, 2, 1, 4, 1, 4, 1, 2, 1, 6, 1, 4, 1, 2, 1, 3, 1

Offset: 0

Eric W. Weisstein, Apr 06 2006

a(0), a(2) and a(8) are the conjectured to be the only terms equal to 0.

			For n=0, 2^0 = 1 is also "1" in base-3, thus there are no 2-digits present, and therefore a(0) = 0.
For n=4, 2^4 = 16, which in base-3 is "121" as 1*(3^2) + 2*3 + 1 = 16, so the rightmost 2 occurs at two steps from the end, therefore a(4) = 2.
For n=5, 2^5 = 32, which in base-3 is "1012" as 1*(3^3) + 1*3 + 2*1 = 32, so the rightmost 2 occurs as the least significant digit (which is the position 1), therefore a(5) = 1.

Antti Karttunen, Table of n, a(n) for n = 0..16384
Eric Weisstein's World of Mathematics, Ternary

Cf. A004642 (ternary expansion of 2^n), A007089. See also A117970.

pf2[n_]:=Module[{p=Position[Reverse[IntegerDigits[2^n,3]],2,{1},1]},If[p=={},0,p]]; Flatten[Array[pf2,110]] (* Harvey P. Dale, Nov 30 2013 *)

A117971(n) = { my(n=(2^n),i=1); while(n, if(2==(n%3),return(i)); i++; n \= 3); (0); }; \\ Antti Karttunen, Mar 30 2021

Edited by Charles R Greathouse IV, Aug 05 2010

Term a(0) = 0 prepended, examples added and the definition clarified by Antti Karttunen, Mar 30 2021

A260683 Number of 2's in the expansion of 2^n in base 3.

Original entry on oeis.org

0, 1, 0, 2, 1, 1, 1, 2, 0, 4, 2, 4, 3, 3, 2, 6, 5, 5, 3, 7, 4, 7, 5, 4, 1, 5, 2, 8, 8, 7, 9, 9, 8, 7, 7, 8, 4, 6, 8, 9, 11, 11, 7, 11, 10, 8, 9, 8, 8, 10, 11, 16, 13, 10, 9, 12, 13, 16, 12, 13, 15, 15, 11, 15, 16, 14, 14, 12, 14, 15, 14, 16, 11, 18, 11, 17, 10

Offset: 0

Emmanuel Vantieghem, Nov 15 2015

Erdős conjectures that a(n) > 0 for n > 8.

			For n=5, the expansion of 2^n in number base 3 is 1012, thus: a(n)=1
For n=10, the expansion of 2^n in number base 3 is 1101221, thus: a(n)=2

R. K. Guy, Unsolved Problems in Number Theory, B33. [Does not seem to be in section B33.]

Robert Israel, Table of n, a(n) for n = 0..10000
Paul Erdős, Some unconventional problems in number theory, Mathematics Magazine, Vol. 52, No. 2 (1979), pp. 67-70.

Cf. A004642 (2^n in base 3), A020915 (number of terms), A036461 (number of 1's), A104320 (number of 0's).
Cf. A000108 (conjecture that A000108(n) is 6m+1 only for n = 0, 1 and 5 follows from Erdős's one).
Cf. A005836 (for numbers with no 2 in base 3).

seq(numboccur(2, convert(2^n,base,3)),n=0..100); # Robert Israel, Nov 15 2015

S={};n=-1;While[n<150,n++;A=IntegerDigits[2^n,3];k=Count[A,2];AppendTo[S, k]];S

c(k, d, b) = {my(c=0, f); while (k>b-1, f=k-b*(k\b); if (f==d, c++); k\=b); if (k==d, c++); return(c)}
for(n=0, 300, print1(c(2^n, 2, 3)", ")) \\ Altug Alkan, Nov 15 2015

a(n) = #select(x->(x==2), digits(2^n, 3)); \\ Michel Marcus, Nov 28 2018

a(n) = hammingweight(digits(2^n, 3)\2); \\ Ruud H.G. van Tol, May 09 2024

use ntheory ":all"; sub a260683 { scalar grep { $==2 } todigits(vecprod((2) x shift), 3) } # _Dana Jacobsen, Aug 16 2016

a(n) = A020915(n) - A104320(n) - A036461(n). - Altug Alkan, Nov 15 2015

a(n) = A081603(A000079(n)). - Michel Marcus, Dec 03 2015

A265210 Irregular triangle read by rows in which row n lists the base 3 digits of 2^n in reverse order, n >= 0.

Original entry on oeis.org

1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 0, 1, 1, 0, 1, 2, 2, 0, 2, 1, 1, 1, 1, 1, 0, 0, 1, 2, 2, 2, 0, 0, 2, 1, 2, 2, 1, 0, 1, 1, 2, 1, 2, 0, 1, 2, 2, 1, 0, 2, 1, 2, 1, 2, 1, 2, 0, 1, 0, 2, 0, 2, 0, 1, 1, 1, 2, 0, 1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1

Offset: 0

L. Edson Jeffery, Dec 04 2015

The length of row n is A020915(n) = 1 + A136409(n).

Conjecture 1: The sequence in column k is periodic, with period p(k) = 2*3^(k-1) = A008776(k-1), k >= 1, and in which the numbers 0,1,2 appear with equal frequency, for each k>1.

			n
0:    1
1:    2
2:    1  1
3:    2  2
4:    1  2  1
5:    2  1  0  1
6:    1  0  1  2
7:    2  0  2  1  1
8:    1  1  1  0  0  1
9:    2  2  2  0  0  2
10:   1  2  2  1  0  1  1
11:   2  1  2  0  1  2  2
12:   1  0  2  1  2  1  2  1
13:   2  0  1  0  2  0  2  0  1
14:   1  1  2  0  1  1  1  1  2
15:   2  2  1  1  2  2  2  2  1  1

Cf. A000079 (powers of 2), A004642 (powers of 2 written in base 3), A008776 (2*3^n).
Cf. A265209 (base 3 digits of 2^n).
Cf. A264980 (row n read as ternary number).
Cf. A037096 (numbers constructed from the inverse case, base 2 digits of 3^n).

(* Replace Flatten with Grid to display the triangle: *)
Flatten[Table[Reverse[IntegerDigits[2^n, 3]], {n, 0, 15}]]

A265210_row(n)=Vecrev(digits(2^n,3)) \\ M. F. Hasler, Dec 05 2015

A346497 List of powers of 2 written in base 3 which contain no zero digits.

Original entry on oeis.org

1, 2, 11, 22, 121, 1122221122

Offset: 1

Rafael Castro Couto, Jul 20 2021

The listed terms are the base-3 expansions of 1, 2, 4, 8, 16, and 32768.
The program shows that there are no other terms less than 2^1000.
a(7) > 2^(10^7). - Martin Ehrenstein, Jul 27 2021
If it exists, a(7) > 2^(10^21). - Robert Saye, Mar 23 2022

David Wells, "The Penguin Dictionary of Curious and Interesting Numbers" (1997), p. 123.

R. C. Couto, Number of nonzero digits in 2^n base 3
Robert I. Saye, On two conjectures concerning the ternary digits of powers of two, arXiv:2202.13256 [math.NT], 2022; J. Integer Seq. 25 (2022) Article 22.3.4.

Cf. A102483, A004642 (all powers of 2 in base 3), A104320 (number of zeros in ternary representation of 2^n), A130693 (same problem in base 10).

pwr = 1; Do[pwr = Mod[2*pwr, 3^100]; d = Union[IntegerDigits[pwr, 3]]; If[Intersection[d, {0}] == {}, Print[IntegerString[pwr, 3]]], {n, 10000000}] (* Ricardo Bittencourt, Jul 07 2021 *)
Select[Table[FromDigits[IntegerDigits[2^n,3]],{n,0,100}],DigitCount[#,10,0]==0&] (* Harvey P. Dale, Feb 18 2025 *)

a(n) = A007089(2^A102483(n)). - Michel Marcus, Jul 23 2021

A364049 a(n) is the least k such that the base-n digits of 2^k are not all distinct.

Original entry on oeis.org

2, 2, 4, 5, 6, 3, 6, 11, 16, 14, 11, 12, 8, 4, 8, 15, 16, 12, 16, 18, 9, 17, 15, 14, 24, 13, 16, 15, 10, 5, 10, 19, 24, 14, 21, 15, 18, 15, 19, 17, 17, 28, 18, 12, 24, 23, 31, 24, 31, 20, 26, 44, 35, 33, 25, 18, 36, 14, 14, 18, 12, 6, 12, 23, 45, 37, 38, 24, 20, 35, 36, 26, 51, 31, 33, 47, 34, 34

Offset: 2

Robert Israel, Jul 03 2023

			a(10) = 16 because 2^16 = 65536 does not have all distinct digits in base 10, while 2^k does have all distinct digits for 1 <= k <= 15.

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A004642, A004643, A000866, A004645, A004646, A004647, A001357, A000079, A011557, A364052, A364089.

f:= proc(n) local k,L;
  for k from 2 do
    L:= convert(2^k,base,n);
    if nops(L) <> nops(convert(L,set)) then return k fi
  od;
end proc:
map(f, [$2..100]);

from itertools import count
from sympy.ntheory import digits
def a(n): return next(k for k in count(2) if len(set(d:=digits(1<Michael S. Branicky, Jul 05 2023

A265209 Irregular triangle read by rows in which row n lists the base-3 digits of 2^n, n >= 0.

Original entry on oeis.org

1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 0, 1, 2, 2, 1, 0, 1, 1, 1, 2, 0, 2, 1, 0, 0, 1, 1, 1, 2, 0, 0, 2, 2, 2, 1, 1, 0, 1, 2, 2, 1, 2, 2, 1, 0, 2, 1, 2, 1, 2, 1, 2, 1, 2, 0, 1, 1, 0, 2, 0, 2, 0, 1, 0, 2, 2, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2

Offset: 0

L. Edson Jeffery, Dec 04 2015

The length of row n is A020915(n) = 1 + A136409(n).

			Triangle begins:
  1
  2
  1  1
  2  2
  1  2  1
  1  0  1  2
  2  1  0  1
  1  1  2  0  2
  1  0  0  1  1  1
  2  0  0  2  2  2
  1  1  0  1  2  2  1
  2  2  1  0  2  1  2
  1  2  1  2  1  2  0  1
  1  0  2  0  2  0  1  0  2
  2  1  1  1  1  0  2  1  1
  1  1  2  2  2  2  1  1  2  2

Cf. A000079 (powers of 2), A003137, A004642 (powers of 2 written in base 3).

Cf. A265210 (base 3 digits of 2^n in reverse order).

(* Replace Flatten with Grid to display the triangle: *)
Flatten[Table[IntegerDigits[2^n, 3], {n, 0, 15}]]

for(n=0,15,for(k=1,#digits(2^n,3),print1(digits(2^n,3)[k],", "))) \\ Derek Orr, Dec 24 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

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

Original entry on oeis.org

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

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).

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.

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.

Cf. A004642, A117971, A351927.

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}]

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

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

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

cp's OEIS Frontend

A117970 Position of first 0 counting from the least significant digit in the ternary expansion of 2^n.

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A004645 Powers of 2 written in base 6.

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Magma

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A117971 One-based position of the first 2 from the least significant digit in the ternary expansion of 2^n, or 0 if there are no 2's present.

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A260683 Number of 2's in the expansion of 2^n in base 3.

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A265210 Irregular triangle read by rows in which row n lists the base 3 digits of 2^n in reverse order, n >= 0.

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Mathematica

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A346497 List of powers of 2 written in base 3 which contain no zero digits.

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A364049 a(n) is the least k such that the base-n digits of 2^k are not all distinct.

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