A035064 -id:A035064 - 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

A030705 Numbers k such that the decimal expansion of 9^k contains no zeros (probably finite).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 12, 13, 14, 17, 34

Offset: 1

Eric W. Weisstein

Integers in A030700 / 2. - M. F. Hasler, Mar 07 2014

No more terms <= 10^5. - Georg Fischer, Mar 12 2020

Eric Weisstein's World of Mathematics, Zero

Cf. A007377 (analog for 2^n), A030700 (3), A030701 (4), A008839 (5), A030702 (6), A030703 and A195908 (7), A030704 (8), A030706 and A195946 (11), A195944 and A195945 (13); A195942, A195943.
This is row 0 of A305929.
Cf. A035064.

[n: n in [0..500] | not 0 in Intseq(9^n)]; // Vincenzo Librandi, Mar 08 2014

Reap[For[n = 0, n < 100, n++, If[FreeQ[IntegerDigits[9^n], 0], Sow[n]]]][[2, 1]] (* Jean-François Alcover, Oct 04 2017 *)

select( is(n)=vecmin(digits(9^n)), [0..39]) \\ M. F. Hasler, Mar 07 2014

Offset changed to 1 and initial 0 added by M. F. Hasler, Mar 07 2014

Removed keyword "fini" (as in A035064) since it is only a conjecture that this sequence contains only finitely many terms. - Georg Fischer, Mar 12 2020

A065744 Number of 9's in the decimal expansion of 2^n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 3, 3, 2, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 0, 0, 0, 1, 3, 3, 2, 3, 5, 3, 3, 3, 0, 0, 0, 2, 3, 1, 1, 1, 2, 2, 3, 3, 2, 1, 0, 2, 5, 2, 3, 2, 0, 2, 2, 3, 3, 1, 4, 3, 2, 1, 2, 2, 4, 4, 2, 6, 8, 3, 3, 3, 1, 1, 0, 2

Offset: 0

Benoit Cloitre, Dec 04 2001

See A035064 for the indices of zeros. I conjecture that any value x = 0, 1, 2, ... occurs only a finite number of times N(x) = 37, 27, 36, 46, 20, 31, 32, 30, 46, 29, 22, ... in this sequence, for the last time at well defined indices i(x) = 108, 197, 296, 277, 278, 315, 379, 555, 503, 504, 539, 696, 667, ... - M. F. Hasler, Jul 09 2025

			2^12 = 4096 so a(12)=1.

Harry J. Smith, Table of n, a(n) for n = 0..1000

Similar for other digits: A027870 (0's), A065712 (1's), A065710 (2's), A065714 (3's), A065715 (4's), A065716 (5's), A065717 (6's), A065718 (7's), A065719 (8's).

Cf. A035064 (2^n has no digit 9).

Table[ Count[ IntegerDigits[2^n], 9], {n, 0, 100} ]

Count(x,d)={ #select(t->t==d, digits(x)) }
a(n) = Count(2^n, 9) \\ Harry J. Smith, Oct 27 2009

def A065744(n):
    return str(2**n).count('9')  # Chai Wah Wu, Feb 14 2020

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

A034293 Numbers k such that 2^k does not contain the digit 2 (probably finite).

Original entry on oeis.org

0, 2, 3, 4, 6, 12, 14, 16, 20, 22, 23, 26, 34, 35, 36, 39, 42, 46, 54, 64, 74, 83, 168

Offset: 1

Erich Friedman

Is 168 the last term?
First row of A136291. - R. J. Mathar Apr 29 2008
Equivalently, indices of zeros in A065710. - M. F. Hasler, Feb 10 2023

			Here is 2^168, conjecturally the largest power of 2 that does not contain a 2: 374144419156711147060143317175368453031918731001856. - _N. J. A. Sloane_, Feb 10 2023

Cf. A007377.
See also similar sequences listed in A035064.
Cf. A065710 (number of '2's in 2^n), A094776.

[n: n in [0..1000] | not 2 in Intseq(2^n) ]; // Vincenzo Librandi, May 07 2015

isA034293 := proc(n) RETURN(not 2 in convert(2^n,base,10)) ; end: for n from 0 to 100000 do if isA034293(n) then print(n) ; fi ; od: # R. J. Mathar, Oct 04 2007

Join[{0}, Select[ Range@10000, FreeQ[ IntegerDigits[2^# ], 2] &]] (* Shyam Sunder Gupta, Sep 01 2007 *)(* adapted by Vincenzo Librandi, May 07 2015 *)
Select[Range[0, 10^4], DigitCount[2^#][[2]] == 0 &] (* Michael De Vlieger, Apr 29 2016 *)

is(n)=setsearch(Set(digits(2^n)),2)==0 \\ Charles R Greathouse IV, May 10 2016

is_A034293(n)=!foreach(digits(2^n),d,d==2&&return) \\ M. F. Hasler, Feb 10 2023

def is_A034293(n): return'2'not in str(2**n)
[n for n in range(199) if is_A034293(n)] # M. F. Hasler, Feb 10 2023

The last term is A094776(2), by definition. - M. F. Hasler, Feb 10 2023

Edited by N. J. A. Sloane, Oct 03 2007

Removed keyword "fini" since it is only a conjecture that this sequence contains only finitely many terms. - Altug Alkan, May 07 2016

A035057 Numbers k such that 2^k does not contain the digit 1 (probably finite).

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 11, 12, 15, 16, 19, 23, 25, 28, 32, 33, 35, 38, 43, 52, 56, 59, 63, 66, 73, 91

Offset: 1

Patrick De Geest, Nov 15 1998

No other terms < 10^6. - Chai Wah Wu, Aug 31 2017

Cf. similar sequences listed in A035064.

[n: n in [0..1000] | not 1 in Intseq(2^n) ]; // Vincenzo Librandi, May 06 2015

Select[Range[100],DigitCount[2^#,10,1]==0&] (* Harvey P. Dale, Jun 06 2013 *)

A035057=[n|n<-[0..99], !setsearch(Set(digits(2^n)),1)] \\ M. F. Hasler, Jul 09 2025

A035057_list = [n for n in range(1,10**3) if '1' not in str(2**n)] # Chai Wah Wu, Aug 31 2017

A035058 Numbers k such that 2^k does not contain the digit 3 (probably finite).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 18, 19, 20, 21, 24, 26, 32, 34, 38, 40, 44, 48, 50, 53, 57, 60, 80, 91, 92, 102, 153

Offset: 1

Patrick De Geest, Nov 15 1998

No more terms <= 10^4. - Georg Fischer, Mar 12 2020

Next term > 2*10^5. - Tyler NeSmith, Jul 30 2021

Cf. similar sequences listed in A035064.

[n: n in [0..1000] | not 3 in Intseq(2^n) ]; // Vincenzo Librandi, May 07 2015

Join[{0}, Select[Range@10000, FreeQ[IntegerDigits[2^#], 3] &]] (* Vincenzo Librandi, May 07 2015 *)

isok(n) = ! vecsearch(Set(digits(2^n)), 3); \\ Michel Marcus, Feb 09 2018

Initial 0 added by Vincenzo Librandi, May 07 2015

Removed keyword "fini" as in A035064, since it is only a conjecture that this sequence contains only finitely many terms. - Georg Fischer, Mar 12 2020

A035059 Numbers k such that 2^k does not contain the digit 4 (probably finite).

Original entry on oeis.org

0, 1, 3, 4, 5, 7, 8, 9, 13, 15, 16, 17, 21, 23, 24, 29, 40, 41, 43, 55, 69, 75, 85, 107

Offset: 1

Patrick De Geest, Nov 15 1998

a(25) > 2*10^9 if it exists. - Jon E. Schoenfield and Michael S. Branicky, Aug 02 2021

Cf. similar sequences listed in A035064.

[n: n in [0..1000] | not 4 in Intseq(2^n) ]; // Vincenzo Librandi, May 07 2015

Join[{0}, Select[Range@10000, FreeQ[IntegerDigits[2^#], 4] &]] (* Vincenzo Librandi, May 07 2015 *)

N = 200; modder = 10**N; REPORT = 10**5
def ok(s):  return '4' not in s
def ok1(n): return ok(str(pow(2, n, modder)))
def afind(limit, startk=0, verbose=False):
    full_tests = 0
    for k in range(startk, limit):
        if ok1(k):
            full_tests += 1
            if ok(str(pow(2, k))): print(k, end=", ")
        if verbose and k and k%REPORT == 0:
            print("[ ...", k, full_tests, "]")
        k += 1
afind(10**6, verbose=True) # Michael S. Branicky, Aug 02 2021

Initial 0 inserted by Vincenzo Librandi, May 07 2015

A035060 Numbers k such that 2^k does not contain the digit 5 (probably finite).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 22, 23, 24, 26, 27, 30, 31, 32, 34, 36, 38, 43, 46, 55, 62, 65, 66, 71

Offset: 1

Patrick De Geest, Nov 15 1998

Cf. similar sequences listed in A035064.

[n: n in [0..1000] | not 5 in Intseq(2^n) ]; // Vincenzo Librandi, May 07 2015

Join[{0}, Select[Range@10000, FreeQ[IntegerDigits[2^#], 5] &]] (* Vincenzo Librandi, May 07 2015 *)
Select[Range[0,80],DigitCount[2^#,10,5]==0&] (* Harvey P. Dale, Dec 24 2018 *)

Initial 0 inserted by Vincenzo Librandi, May 07 2015

A035061 Numbers n such that 2^n does not contain the digit 6 (probably finite).

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 9, 10, 11, 13, 17, 19, 21, 22, 25, 27, 30, 33, 37, 39, 41, 45, 47, 53, 54, 57, 90, 93

Offset: 1

Patrick De Geest, Nov 15 1998

Cf. similar sequences listed in A035064.

[n: n in [0..1000] | not 6 in Intseq(2^n) ]; // Vincenzo Librandi, May 07 2015

Join[{0}, Select[Range@10000, FreeQ[IntegerDigits[2^#], 6] &]] (* Vincenzo Librandi, May 07 2015 *)

Initial 0 added by Vincenzo Librandi, May 07 2015

A035062 Numbers n such that 2^n does not contain the digit 7.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 18, 19, 22, 23, 25, 28, 33, 41, 42, 49, 50, 54, 61, 71

Offset: 1

Patrick De Geest, Nov 15 1998

It is conjectured that 71 is the last term.

Cf. similar sequences listed in A035064.

[n: n in [0..1000] | not 7 in Intseq(2^n) ]; // Vincenzo Librandi, May 06 2015

Join[{0}, Select[ Range@10000, FreeQ[ IntegerDigits[2^# ], 7] &]] (* Shyam Sunder Gupta, Sep 01 2007 *)

is(n)=!#select(k->k==7, Set(digits(2^n))) \\ Charles R Greathouse IV, May 06 2015

Edited by N. J. A. Sloane, May 21 2008 at the suggestion of R. J. Mathar

Initial 0 added and Mathematica code adapted by Vincenzo Librandi, May 06 2015

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

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A030705 Numbers k such that the decimal expansion of 9^k contains no zeros (probably finite).

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A065744 Number of 9's in the decimal expansion of 2^n.

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A034293 Numbers k such that 2^k does not contain the digit 2 (probably finite).

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A035057 Numbers k such that 2^k does not contain the digit 1 (probably finite).

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A035058 Numbers k such that 2^k does not contain the digit 3 (probably finite).

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Mathematica