A102483 -id:A102483 - cp's OEIS frontend

A238936 Powers of 6 without the digit '0' in their decimal expansion.

1, 6, 36, 216, 1296, 7776, 46656, 279936, 1679616, 2176782336, 16926659444736, 4738381338321616896, 36845653286788892983296, 17324272922341479351919144385642496

Offset: 1

M. F. Hasler, Mar 07 2014

Conjectured to be finite and complete. See the OEIS wiki page for further information, references and links.

M. F. Hasler, Zeroless powers, OEIS wiki, Mar 07 2014

Cf. A007377, A030700, A030701, A008839, A030702, A030703, A030705, A030706, A195944.
Cf. A195908, A195946, A195945.
Cf. A195942, A195943, A027870, A102483.

Select[6^Range[0,50],DigitCount[#,10,0]==0&] (* Harvey P. Dale, Dec 03 2020 *)

for(n=0,99,vecmin(digits(6^n))&& print1(6^n","))

a(n)=6^A030702(n).

Keyword:fini and keyword:full removed by Jianing Song, Jan 28 2023 as finiteness is only conjectured.

A238940 Powers of 4 without the digit '0' in their decimal expansion.

Original entry on oeis.org

1, 4, 16, 64, 256, 16384, 65536, 262144, 16777216, 268435456, 4294967296, 17179869184, 68719476736, 4722366482869645213696, 75557863725914323419136, 77371252455336267181195264

Offset: 1

M. F. Hasler, Mar 07 2014

Conjectured to be finite and complete. See the OEIS wiki page for further information, references and links.

M. F. Hasler, Zeroless powers, OEIS Wiki, Mar 07 2014

For the zeroless numbers (powers x^n), see A238938, A238939, A238940, A195948, A238936, A195908, A195946, A195945, A195942, A195943.
For the corresponding exponents, see A007377, A008839, A030700, A030701, A030702, A030703, A030704, A030705, A030706, A195944.
For other related sequences, see A052382, A027870, A102483.

Select[4^Range[0,50],DigitCount[#,10,0]==0&] (* Harvey P. Dale, Aug 31 2021 *)

for(n=0,99,vecmin(digits(4^n))&& print1(4^n","))

a(n)=4^A030701(n).

Keyword:fini removed by Jianing Song, Jan 28 2023 as finiteness is only conjectured.

A306112 Largest k such that 2^k has exactly n digits 0 (in base 10), conjectured.

Original entry on oeis.org

86, 229, 231, 359, 283, 357, 475, 476, 649, 733, 648, 696, 824, 634, 732, 890, 895, 848, 823, 929, 1092, 1091, 1239, 1201, 1224, 1210, 1141, 1339, 1240, 1282, 1395, 1449, 1416, 1408, 1616, 1524, 1727, 1725, 1553, 1942, 1907, 1945, 1870, 1724, 1972, 1965, 2075, 1983, 2114, 2257, 2256

Offset: 0

M. F. Hasler, Jun 22 2018

a(0) is the largest term in A007377: exponents of powers of 2 without digit 0.

There is no proof for any of the terms, just as for any term of A020665 and many similar / related sequences. However, the search has been pushed to many magnitudes beyond the largest known term, and the probability of any of the terms being wrong is extremely small, cf., e.g., the Khovanova link.

M. F. Hasler, Zeroless powers, OEIS Wiki, March 2014, updated 2018.
T. Khovanova, The 86-conjecture, Tanya Khovanova's Math Blog, Feb. 2011.
W. Schneider, No Zeros, 2000, updated 2003. (On web.archive.org--see A007496 for a cached copy.)

Cf. A031146: least k such that 2^k has n digits 0 in base 10.
Cf. A305942: number of k's such that 2^k has n digits 0.
Cf. A305932: row n lists exponents of 2^k with n digits 0.
Cf. A007377: { k | 2^k has no digit 0 } : row 0 of the above.
Cf. A238938: { 2^k having no digit 0 }.
Cf. A027870: number of 0's in 2^n (and A065712, A065710, A065714, A065715, A065716, A065717, A065718, A065719, A065744 for digits 1 .. 9).
Cf. A102483: 2^n contains no 0 in base 3.

A306112_vec(nMax,M=99*nMax+199,x=2,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(x^k)),nMax)]=k);a[^-1]}

A104320 Number of zeros in ternary representation of 2^n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 1, 1, 1, 4, 1, 0, 4, 2, 3, 3, 3, 3, 3, 7, 7, 9, 5, 6, 6, 4, 4, 3, 5, 6, 7, 9, 9, 10, 6, 6, 9, 9, 8, 9, 8, 7, 13, 12, 13, 9, 5, 9, 8, 6, 16, 13, 9, 10, 11, 11, 7, 14, 13, 13, 9, 12, 14, 15, 15, 11, 11, 17, 15, 19, 14, 19, 12, 18, 15, 11, 10, 16, 15, 14, 14, 13, 17, 14

Offset: 0

Reinhard Zumkeller, Mar 01 2005

Conjecture from N. J. A. Sloane: a(n) > 0 for n > 15, see A102483.

			n=13: 2^13=8192 -> '102020102', a(13) = 4.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A004642, A007089.

[Multiplicity(Intseq(2^n,3),0):n in [0..90]]; // Marius A. Burtea, Nov 17 2019

f:= n -> numboccur(0, convert(2^n,base,3)):
map(f, [$0..100]); # Robert Israel, Nov 17 2019

Table[DigitCount[2^n,3,0],{n,0,90}] (* Harvey P. Dale, May 06 2014 *)

a(n) = my(d=vecsort(digits(2^n, 3))); #setintersect(d, vector(#d)) \\ Felix Fröhlich, Nov 17 2019

a(n) = #select(d->!d, digits(2^n, 3)); \\ Ruud H.G. van Tol, May 09 2024

a(n) = A077267(A000079(n)).

a(A104321(n))=n and a(m)<>n for m < A104321(n).

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

Original entry on oeis.org

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

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

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

A252482 Exponents n such that the decimal expansion of the power 12^n contains no zeros.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 8, 10, 14, 20, 26

Offset: 1

M. F. Hasler, Dec 17 2014

Conjectured to be finite.

See A245853 for the actual powers 12^a(n).

M. F. Hasler, Zeroless powers, OEIS Wiki, Mar 07 2014
Eric Weisstein's World of Mathematics, Zero

For zeroless powers x^n, see A238938 (x=2), A238939, A238940, A195948, A238936, A195908, A245852, A240945 (k=9), A195946 (x=11), A245853, A195945; A195942, A195943, A103662.
For the corresponding exponents, see A007377, A030700, A030701, A008839, A030702, A030703, A030704, A030705, A030706, this sequence A252482, A195944.
For other related sequences, see A052382, A027870, A102483, A103663.

Select[Range[0,30],DigitCount[12^#,10,0]==0&] (* Harvey P. Dale, Apr 06 2019 *)

for(n=0,9e9,vecmin(digits(12^n))&&print1(n","))

cp's OEIS Frontend

A238936 Powers of 6 without the digit '0' in their decimal expansion.

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A238940 Powers of 4 without the digit '0' in their decimal expansion.

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A306112 Largest k such that 2^k has exactly n digits 0 (in base 10), conjectured.

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A104320 Number of zeros in ternary representation of 2^n.

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

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

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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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