A027870 -id:A027870 - cp's OEIS frontend

A007377 Numbers k such that the decimal expansion of 2^k contains no 0.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 14, 15, 16, 18, 19, 24, 25, 27, 28, 31, 32, 33, 34, 35, 36, 37, 39, 49, 51, 67, 72, 76, 77, 81, 86

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

It is an open problem of long standing to show that 86 is the last term.
A027870(a(n)) = A224782(a(n)) = 0. - Reinhard Zumkeller, Apr 30 2013
See A030700 for the analog for 3^k, which seems to end with k=68. - M. F. Hasler, Mar 07 2014
Checked up to k = 10^10. - David Radcliffe, Aug 21 2022

			Here is 2^86, conjecturally the largest power of 2 not containing a 0: 77371252455336267181195264. - _N. J. A. Sloane_, Feb 10 2023

J. S. Madachy, Mathematics on Vacation, Scribner's, NY, 1966, p. 126.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

W. Schneider, NoZeros: Powers n^k without Digit Zero [Cached copy]
Eric Weisstein's World of Mathematics, Zero
R. G. Wilson, V, Letter to N. J. A. Sloane, Oct. 1993

Cf. A027870, A030700, A102483, A034293.
Some similar sequences are listed in A035064.
Cf. also A031142.

import Data.List (elemIndices)
a007377 n = a007377_list !! (n-1)
a007377_list = elemIndices 0 a027870_list
-- Reinhard Zumkeller, Apr 30 2013

[ n: n in [0..50000] | not 0 in Intseq(2^n) ];  // Bruno Berselli, Jun 08 2011

remove(t -> has(convert(2^t,base,10),0),[$0..1000]); # Robert Israel, Dec 29 2015

Do[ If[ Union[ RealDigits[ 2^n ] [[1]]] [[1]] != 0, Print[ n ] ], {n, 1, 60000}]
Select[Range@1000, First@Union@IntegerDigits[2^# ] != 0 &]
Select[Range[0,100],DigitCount[2^#,10,0]==0&] (* Harvey P. Dale, Feb 06 2015 *)

for(n=0,99,if(vecmin(eval(Vec(Str(2^n)))),print1(n", "))) \\ Charles R Greathouse IV, Jun 30 2011

use bignum;
for(0..99) {
  if((1<<$_) =~ /^[1-9]+$/) {
    print "$_, "
  }
} # Charles R Greathouse IV, Jun 30 2011

def ok(n): return '0' not in str(2**n)
print(list(filter(ok, range(10**4)))) # Michael S. Branicky, Aug 08 2021

a(1) = 0 prepended by Reinhard Zumkeller, Apr 30 2013

A020665 a(n) is the (conjectured) maximal exponent k such that n^k does not contain a digit zero in its decimal expansion.

Original entry on oeis.org

86, 68, 43, 58, 44, 35, 27, 34, 0, 41, 26, 14, 34, 27, 19, 27, 17, 44, 0, 13, 22, 10, 13, 29, 15, 9, 16, 14, 0, 16, 7, 23, 5, 17, 22, 16, 10, 19, 0, 9, 13, 10, 6, 39, 7, 8, 19, 5, 0, 19, 18, 7, 13, 11, 23, 7, 23, 14, 0, 16, 5, 14, 12, 3, 14, 14, 14, 12, 0, 8, 22, 6, 4, 19, 11, 12, 10, 9, 0

Offset: 2

David W. Wilson

Most of these values are not proved rigorously, but the search has been pushed very large (~ 10^9 or beyond for many n). See the OEIS wiki page for further reading. - M. F. Hasler, Mar 08 2014
From Bill McEachen, Apr 01 2015: (Start)
It appears that the values at square pointers will be no more than that of the base pointer. Specifically when the value at the base pointer is even, the value at the square will be 50%. For example, the sequence n=2,4,16 yields a(n)=86,43,19. The sequence n=3,9,81 yields a(n)=68,34,17.
Values at other than squares are less obvious. However, at some point, the run of the squares ends, implying remaining nonzero values should indicate either nonsquares or prime entries. (End)
Since (n^b)^j = n^(b*j), a(n) >= b*a(n^b); if a(n) is divisible by b then a(n^b) = a(n)/b. - Robert Israel, Apr 01 2015

			a(13) = 14 because 13^14 does not have a digit 0, but (it is conjectured that) for all k > 14, 13^k will have a digit 0. It is not excluded that there may be some k < a(n) for which n^k does have a digit 0, as is the case for 13^6. - _M. F. Hasler_, Mar 29 2015

Antoine Beaulieu, Table of n, a(n) for n = 2..200 (checked with PARI up to k=10^5)
M. F. Hasler, Zeroless powers, OEIS Wiki, Mar 07 2014
Eric Weisstein's World of Mathematics, Zero

For the zeroless numbers (powers x^n), see A238938, A238939, A238940, A195948, A238936, A195908, A195946, A195945, A195942, A195943, A103662.
For the corresponding exponents, see A007377, A008839, A030700, A030701, A008839, A030702, A030703, A030704, A030705, A030706, A195944.
For other related sequences, see A011540, A052382, A027870, A102483, A103663.

f:= proc(n)
  local p;
  if n mod 10 = 0 then return 0 fi;
  for p from 100 by -1 do
    if not has(convert(n^p,base,10),0) then return(p) fi
  od
0
end proc:
seq(f(n),n=2..80); # Robert Israel, Apr 01 2015

a = {}; Do[ If[ Mod[n, 10] == 0, b = 0; Continue]; Do[ If[ Count[ IntegerDigits[n^k], 0 ] == 0, b = k], {k, 1, 200} ]; a = Append[a, b], {n, 2, 81} ];

Nmax(x,L=99,m=0)=for(n=1,L,vecmin(digits(x^n))&&m=n);m \\ L=99 is enough to reproduce the known results, since no value > 86 is known; M. F. Hasler, Mar 08 2014

a(10n) = 0 for any n>0. - M. F. Hasler, Dec 17 2014
a(100n+1) = 0 for any n>0. - Robert Israel, Apr 01 2015
a(80*n+65) <= 3, because for k >= 4, (80*n+65)^k == 625 (mod 10000). - Robert Israel, Apr 02 2015
From Chai Wah Wu, Jan 08 2020: (Start)
The following values and bounds are for the actual maximal exponents (not conjectured).
a(A052382(n)) > 0 for n > 1.
a(225) = 1
a(225^k) = 0 for k > 1.
a(625) = 1.
a(625^k) = 0 for k > 1.
a(3126) = 2.
a(3126^2) = 1.
a(3126^k) = 0 for k > 2.
a(9376) = 1.
a(9376^k) = 0 for k > 1.
a(21876) = 2.
a(21876^2) = 1.
a(21876^k) = 0 for k > 2.
a(34376) = 1.
a(34376^k) = 0 for k > 1.
a(400*n + 225) <= 1, since for k >= 2, (400*n + 225)^k == 625 (mod 10000), i.e., if 400*n + 225 is in A052382, then a(400*n+225) = 1, otherwise it is 0.
a(25000*n + 34376) <= 1, since for k >= 2, (25000*n + 34376)^k == 9376 (mod 100000), i.e., if 25000*n + 34376 is in A052382, then a(25000*n + 34376) = 1, otherwise it is 0.
a(25000*n + 21876) <= 2, since for k >= 3, (25000*n + 21876)^k == 9376 (mod 100000).
a(12500*n + 3126) <= 4, since for k >= 5, (12500*n + 3126)^k == 9376 (mod 100000).
(End)

A030700 Decimal expansion of 3^n contains no zeros (probably finite).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 19, 23, 24, 26, 27, 28, 31, 34, 68

Offset: 1

Eric W. Weisstein

See A007377 for the analog for 2^n (final term seems to be 86), A008839 for 5^n (final term seems to be 58), and others listed in cross-references. - M. F. Hasler, Mar 07 2014

See A238939(n) = 3^a(n) for the actual powers. - M. F. Hasler, Mar 08 2014

			Here is 3^68, conjecturally the largest power of 3 that does not contain a zero: 278128389443693511257285776231761. - _N. J. A. Sloane_, Feb 10 2023

M. F. Hasler, Zeroless powers, OEIS Wiki, Mar 07 2014
W. Schneider, NoZeros: Powers n^k without Digit Zero [Cached copy]
Eric Weisstein's World of Mathematics, Zero

For the zeroless numbers (powers x^n), see A238938, A238939, A238940, A195948, A238936, A195908, A195946, A195945, A195942, A195943, A103662.
For the corresponding exponents, see A007377, A030700 (this), A030701, A008839, A030702, A030703, A030704, A030705, A030706, A195944.
For other related sequences, see A052382, A027870, A102483, A103663.

[n: n in [0..500] | not 0 in Intseq(3^n) ]; // Vincenzo Librandi, Oct 19 2012

Do[If[Union[RealDigits[3^n][[1]]][[1]]!=0, Print[n]], {n, 0,10000}] (* Vincenzo Librandi, Oct 19 2012 *)
Select[Range[0,70],DigitCount[3^#,10,0]==0&] (* Harvey P. Dale, Feb 06 2019 *)

is_A030700(n)=vecmin(digits(3^n)) \\ M. F. Hasler, Mar 07 2014

A030700=select( is_A030700, [0..199]) \\ M. F. Hasler, Jun 14 2018

Initial term 0 added by Vincenzo Librandi, Oct 19 2012

A008839 Numbers k such that the decimal expansion of 5^k contains no zeros.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 17, 18, 30, 33, 58

Offset: 1

Olivier Gérard

Probably 58 is last term.

Searched for k up to 10^10. - David Radcliffe, Dec 27 2015

			Here is 5^58, conjecturally the largest power of 5 that does not contain a 0:
34694469519536141888238489627838134765625. - _N. J. A. Sloane_, Feb 10 2023, corrected by _Patrick De Geest_, Jun 09 2024

M. F. Hasler, Zeroless powers, OEIS Wiki, Mar 07 2014.
W. Schneider, NoZeros: Powers n^k without Digit Zero [Cached copy]
Eric Weisstein's World of Mathematics, Zero.

Cf. A000351 (5^n).
For the zeroless numbers (powers x^n), see A238938, A238939, A238940, A195948, A238936, A195908, A195946, A195945, A195942, A195943, A103662.
For the corresponding exponents, see A007377, A008839, A030700, A030701, A008839, A030702, A030703, A030704, A030705, A030706, A195944.
For other related sequences, see A305925, A052382, A027870, A102483, A103663.

[ n: n in [0..500] | not 0 in Intseq(5^n) ]; // Vincenzo Librandi Oct 19 2012

Do[ If[ Union[ RealDigits[ 5^n ][[1]]] [[1]] != 0, Print[ n ]], {n, 0, 25000}]

for(n=0,99,vecmin(digits(5^n))&& print1(n",")) \\ M. F. Hasler, Mar 07 2014

Definition corrected and initial term 0 added by M. F. Hasler, Sep 25 2011
Further edits by M. F. Hasler, Mar 08 2014
Keyword:fini removed by Jianing Song, Jan 28 2023 as finiteness is only conjectured.

A030701 Decimal expansion of 4^n contains no zeros (probably finite).

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 8, 9, 12, 14, 16, 17, 18, 36, 38, 43

Offset: 1

Eric W. Weisstein

Integers in A007377 / 2. Conjectured to be finite, and probably complete. - M. F. Hasler, Mar 08 2014

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

For the zeroless numbers (powers x^n), see A238938, A238939, A238940, A195948, A238936, A195908, A195946, A195945, A195942, A195943, A103662.
For the corresponding exponents, see A007377, A008839, A030700, A030701, A008839, A030702, A030703, A030704, A030705, A030706, A195944.
For other related sequences, see A052382, A027870, A102483, A103663.

[n: n in [0..500] | not 0 in Intseq(4^n)]; // Vincenzo Librandi, Mar 08 2014

Select[Range[0,50],DigitCount[4^#,10,0]==0&] (* Paolo Xausa, Oct 07 2023 *)

for(n=0, 99, vecmin(digits(4^n))&& print1(n", ")) \\ M. F. Hasler, Mar 07 2014

Offset corrected and initial 0 added by M. F. Hasler, Mar 07 2014

A094776 a(n) = largest k such that the decimal representation of 2^k does not contain the digit n.

Original entry on oeis.org

86, 91, 168, 153, 107, 71, 93, 71, 78, 108

Offset: 0

Michael Taktikos, Jun 09 2004

These values are only conjectural.

The sequence could be extended to any nonnegative integer index n defining a(n) to be the largest k such that n does not appear as substring in the decimal expansion of 2^k. I conjecture that for n = 10, 11, 12, ... it continues (2000, 3020, 1942, 1465, 1859, 2507, 1950, 1849, 1850, ...). For example, curiously enough, the largest power of 2 in which the string "10" does not appear seems to be 2^2000. - M. F. Hasler, Feb 10 2023

			a(0) = 86 because 2^86 = 77371252455336267181195264 is conjectured to be the highest power of 2 that doesn't contain the digit 0.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 71, p. 25, Ellipses, Paris 2008.

Tanya Khovanova, 86 Conjecture, T. K.'s Math Blog, Feb. 15, 2011.
Popular Computing (Calabasas, CA), Two Tables, Vol. 1, (No. 9, Dec 1973), page PC9-16.

Cf. A259081 - A259083.
Cf. A027870 and A065712 - A065744 (number of '0's, ..., '9's in 2^n).
Cf. A034293 (numbers k such that 2^k has no '2').

f[n_] := Block[{a = {}, k = 1}, While[k < 10000, If[ Position[ Union[ IntegerDigits[ 2^k, 10]], n] == {}, AppendTo[a, k]]; k++ ]; a]; Table[ f[n][[ -1]], {n, 0, 9}] (* Robert G. Wilson v, Jun 12 2004 *)

A094776(n,L=10*20^#Str(n))={forstep(k=L, 0, -1, foreach(digits(1<M. F. Hasler, Feb 13 2023

def A094776(n, L=0):
   n = str(n)
   for k in range(L if L else 10*20**len(n), 0, -1):
      if n not in str(2**k): return k # M. F. Hasler, Feb 13 2023

A030706 Decimal expansion of 11^n contains no zeros (probably finite).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 15, 16, 18, 41

Offset: 1

Eric W. Weisstein

See A195946 for the actual powers 11^n. - M. F. Hasler, Dec 17 2014

It appears that 41 is also the largest integer n such that 11^n is not pandigital, cf. A272269. - M. F. Hasler, May 18 2017

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

For other zeroless powers x^n, see A238938, A238939, A238940, A195948, A238936, A195908 (x=7), A245852, A240945 (k=9), A195946 (x=11), A245853 (x=12), A195945 (x=13); A195942, A195943, A103662.
For the corresponding exponents, see A007377, A030700, A030701, A008839, A030702, A030703, A030704, A030705, A030706 (this), A195944.
For other related sequences, see A052382, A027870, A102483, A103663.

Select[Range[0,41],DigitCount[11^#,10,0]==0&] (* Harvey P. Dale, Dec 31 2020 *)

for(n=0,99,vecmin(digits(11^n))&&print1(n",")) \\ M. F. Hasler, Mar 08 2014

Offset corrected and initial term 0 added by M. F. Hasler, Sep 25 2011

Further edits by M. F. Hasler, Dec 17 2014

A195945 Powers of 13 which have no zero in their decimal expansion.

Original entry on oeis.org

1, 13, 169, 2197, 28561, 371293, 62748517, 137858491849, 3937376385699289

Offset: 1

M. F. Hasler, Sep 25 2011

Probably finite. Is 3937376385699289 the largest term?
No further terms up to 13^25000. - Harvey P. Dale, Oct 01 2011
No further terms up to 13^45000. - Vincenzo Librandi, Jul 31 2013
No further terms up to 13^(10^9). - Daniel Starodubtsev, Mar 22 2020

M. F. Hasler, Zeroless powers, OEIS Wiki, Mar 07 2014
C. Rivera, Puzzle 607. A zeroless Prime power, on primepuzzles.net, Sept. 24, 2011.
W. Schneider, NoZeros: Powers n^k without Digit Zero (local copy of www.wschnei.de/digit-related-numbers/nozeros.html), as of Jan 30 2003.

For other zeroless powers x^n, see A238938 (x=2), A238939, A238940, A195948, A238936, A195908, A195946 (x=11), A195945, A195942, A195943, A103662.
For the corresponding exponents, see A007377, A008839, A030700, A030701, A008839, A030702, A030703, A030704, A030705, A030706, A195944 and also A020665.
For other related sequences, see A052382, A027870, A102483, A103663.

[13^n: n in [0..2*10^4] | not 0 in Intseq(13^n)]; // Bruno Berselli, Sep 26 2011

Select[13^Range[0,250],DigitCount[#,10,0]==0&] (* Harvey P. Dale, Oct 01 2011 *)

for(n=0,9999, is_A052382(13^n) && print1(13^n,","))

Equals A001022 intersect A052382 (as a set).

Equals A001022 o A195944 (as a function).

A065710 Number of 2's in the decimal expansion of 2^n.

Original entry on oeis.org

0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 2, 2, 0, 2, 0, 0, 1, 1, 0, 2, 1, 1, 1, 1, 2, 1, 0, 0, 0, 1, 1, 0, 1, 4, 0, 3, 1, 2, 0, 1, 1, 3, 3, 3, 1, 2, 0, 1, 2, 1, 2, 2, 2, 3, 1, 3, 0, 2, 2, 3, 3, 2, 2, 4, 4, 4, 0, 1, 2, 4, 3, 1, 3, 6, 2, 0, 2, 4, 4, 4, 2, 3, 6, 2, 1, 5, 1, 2, 4, 4, 1, 2, 6

Offset: 0

Benoit Cloitre, Dec 04 2001

See A034293 for indices of zeros: It is conjectured that the last 0 appears at index 168 = A094776(2). More generally, I conjecture that any value x = 0, 1, 2, 3, ... occurs only a finite number of times N(x) = 23, 35, 28, 26, 41, 37, 34, 26, 34, 38, 33, 41, ... in this sequence, for the last time at a well defined index i(x) = 168, 176, 186, 268, 423, 361, 472, 555, 470, 562, 563, 735, .... - M. F. Hasler, Feb 10 2023, edited by M. F. Hasler, Jul 09 2025

			2^31 = 2147483648 so a(31) = 1.

M. F. Hasler, Table of n, a(n) for n = 0..10000 (first 1001 terms from Harry J. Smith), Feb 10 2023

Cf. 0's A027870, 1's A065712, 3's A065714, 4's A065715, 5's A065716, 6's A065717, 7's A065718, 8's A065719, 9's A065744.

Cf. A034293, A094776 (largest k for which 2^k has no digit n).

seq(numboccur(2, convert(2^n,base,10)),n=0..100); # Robert Israel, Jul 09 2025

Table[ Count[ IntegerDigits[2^n], 2], {n, 0, 100} ]

a(n) = #select(x->(x==2), digits(2^n)); \\ Michel Marcus, Jun 15 2018

def A065710(n):
    return str(2**n).count('2') # Chai Wah Wu, Feb 14 2020

a(n) = a(floor(n/10)) + [n == 2 (mod 10)], where [...] is the Iverson bracket. - M. F. Hasler, Feb 10 2023

More terms from Robert G. Wilson v, Dec 07 2001

A238938 Powers of 2 without the digit '0' in their decimal expansion.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 8192, 16384, 32768, 65536, 262144, 524288, 16777216, 33554432, 134217728, 268435456, 2147483648, 4294967296, 8589934592, 17179869184, 34359738368, 68719476736, 137438953472, 549755813888, 562949953421312, 2251799813685248, 147573952589676412928

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.

			256 = 2^8 is in the sequence because 256 has a 2, a 5 and a 6 but no 0's.
512 = 2^9 is also in because it has a 1, a 2 and a 5 but no 0's.
1024 = 2^10 is not in the sequence because it has a 0.

Daniel Mondot, Table of n, a(n) for n = 1..36
M. F. Hasler, Zeroless powers, OEIS wiki, Mar 07 2014

Cf. A007377, A030700, A030701, A008839, A030702, A030703, A030704, A030705, A030706, A195944.
Cf. A195908, A195946, A195945.
Cf. A195942, A195943, A027870, A102483.

Select[2^Range[0, 127], DigitCount[#, 10, 0] == 0 &] (* Alonso del Arte, Mar 07 2014 *)

for(n=0,99,vecmin(digits(2^n))&& print1(2^n","))

a(n) = 2^A007377(n).

'fini' keyword removed as finiteness is only conjectured by Max Alekseyev, Apr 10 2019

cp's OEIS Frontend

A007377 Numbers k such that the decimal expansion of 2^k contains no 0.

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A020665 a(n) is the (conjectured) maximal exponent k such that n^k does not contain a digit zero in its decimal expansion.

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A030700 Decimal expansion of 3^n contains no zeros (probably finite).

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A008839 Numbers k such that the decimal expansion of 5^k contains no zeros.

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A030701 Decimal expansion of 4^n contains no zeros (probably finite).

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A094776 a(n) = largest k such that the decimal representation of 2^k does not contain the digit n.

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