A034293 -id:A034293 - 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

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

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

A035064 Numbers k such that 2^k does not contain the digit 9 (probably finite).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 16, 17, 18, 19, 20, 23, 24, 25, 26, 27, 28, 30, 31, 45, 46, 47, 57, 58, 59, 71, 77, 99, 108

Offset: 1

Patrick De Geest, Nov 15 1998

			Here is 2^108, conjecturally the largest power of 2 that does not contain a 9: 324518553658426726783156020576256. - _N. J. A. Sloane_, Feb 10 2023

Cf. numbers n such that decimal expansion of 2^n contains no k: A007377 (k=0), A035057 (k=1), A034293 (k=2), A035058 (k=3), A035059 (k=4), A035060 (k=5), A035061 (k=6), A035062 (k=7), A035063 (k=8), this sequence (k=9).

Indices of zeros in A065744 (number of 9s in digits of 2^n).

[n: n in [0..1000] | not 9 in Intseq(2^n) ]; // Vincenzo Librandi, May 06 2015

Join[{0}, Select[Range@ 1000, FreeQ[IntegerDigits[2^#], 9] &]] (* Vincenzo Librandi, May 06 2015 *)

A035064 = select( is_A035064(n)=vecmax(digits(2^n))<9, [0..199]) \\ M. F. Hasler, Jul 09 2025

(A035064:=[n for n in range(123) if max(str(2**n))<'9']) # M. F. Hasler, Jul 09 2025

Initial 0 added by Vincenzo Librandi, May 06 2015

Removed keyword "fini" at the suggestion of Nathan Fox, since it is only a conjecture that this sequence contains only finitely many terms. - N. J. A. Sloane, Mar 03 2016

A113951 Largest number whose n-th power is exclusionary (or 0 if no such number exists).

Original entry on oeis.org

639172, 7658, 2673, 0, 92, 93, 712, 0, 18, 12, 4, 0, 37, 0, 9, 0, 0, 3, 4, 0, 7, 2, 7, 0, 8, 3, 9, 0, 0, 0, 0, 0, 3, 2, 2, 0, 0, 7, 3, 0, 2, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 2, 3, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3

Offset: 2

Lekraj Beedassy, Nov 09 2005

The number m with no repeated digits has an exclusionary n-th power m^n if the latter is made up of digits not appearing in m. For the corresponding m^n see A113952. In principle, no exclusionary n-th power exists for n == 1 (mod 4) = A016813.

After a(84) = 3, the next nonzero term is a(168) = 2, where 168 is the last known term in A034293. - Michael S. Branicky, Aug 28 2021

			a(4) = 2673 because no number with distinct digits beyond 2673 has a 4th power that shares no digit in common with it (see A111116). Here we have 2673^4 = 51050010415041.

H. Ibstedt, Solution to Problem 2623, "Exclusionary Powers", pp. 346-9 Journal of Recreational Mathematics, Vol. 32 No.4 2003-4, Baywood NY.

Michael S. Branicky, Table of n, a(n) for n = 2..225

Cf. A109135, A112736, A112994, A113318, A034293.

from itertools import combinations, permutations
def no_repeated_digits():
    for d in range(1, 11):
        for p in permutations("0123456789", d):
            if p[0] == '0': continue
            yield int("".join(p))
def a(n):
    m = 0
    for k in no_repeated_digits():
        if set(str(k)) & set(str(k**n)) == set():
            m = max(m, k)
    return m
for n in range(2, 4): print(a(n), end=", ") # Michael S. Branicky, Aug 28 2021

a(34), a(39), a(40) corrected by and a(43) and beyond from Michael S. Branicky, Aug 28 2021

A136291 Array read by rows: each row is a sequence of numbers k such that n^k does not contain the digit n.

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, 0, 2, 3, 4, 6, 7, 8, 10, 11, 14, 19, 27, 28, 34, 40, 50, 55, 84, 0, 2, 4, 8, 12, 20, 0, 0, 0, 2, 3, 4, 7, 16, 22, 24, 39, 0, 2, 3, 4, 6, 7, 8, 26, 0, 2, 4, 6, 8, 10, 16, 28

Offset: 0

Zak Seidov, Mar 20 2008

Last values in each row are 168,84,20,0,0,39,26,28 (all terms highly probably correct).

			Each row is sequence of numbers k such that n^k does not contain the digit n (all rows probably finite, checked up to k=10^4), n=2..9:
0,2,3,4,6,12,14,16,20,22,23,26,34,35,36,39,42,46,54,64,74,83,168 (A034293)
0,2,3,4,6,7,8,10,11,14,19,27,28,34,40,50,55,84 (A131629)
0,2,4,8,12,20
0,
0,
0,2,3,4,7,16,22,24,39
0,2,3,4,6,7,8,26
0,2,4,6,8,10,16,28.

Cf. A034293, A131629.

A185186 Numbers divisible by at least one of their digits other than 1.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 12, 15, 20, 22, 24, 25, 26, 28, 30, 32, 33, 35, 36, 39, 40, 42, 44, 45, 48, 50, 52, 55, 60, 62, 63, 64, 65, 66, 70, 72, 75, 77, 80, 82, 84, 85, 88, 90, 92, 93, 95, 96, 99, 102, 104, 105, 112, 115, 120, 122, 123, 124, 125, 126, 128, 132

Offset: 1

Zak Seidov, Mar 11 2011

The only primes in the sequence are 2, 3, 5, 7. No repunits are eligible.
Also, an interesting class of non-eligible integers consists of some powers of 2, 3 and 7:
"2, 4, 8-less" powers of 2, 2^m = 1, 16, 65536 with m = 0, 4, 16 (a subsequence of A034293);
"3, 9-less" powers of 3, 3^m = {1, 27, 81, 177147, 1162261467}, with m = {0, 3, 4, 11, 19} (a subsequence of A131629);
"seven-less" powers of 7, 7^m, with m = 0, 2, 3, 4, 7, 16, 22, 24, 39 (see 6th row of A136291 Array read by rows: each row is a sequence of numbers k such that n^k does not contain the digit n).
Asymptotic density 27/35 = 0.771... - Charles R Greathouse IV, Mar 11 2011
The asymptotic density of numbers having a prime digit is 1 for each prime digit. The asymptotic density of numbers being divisible by 2, 3, 5 or 7 is -Sum_{d|210, d>1}((-1)^omega(d) / d) = 27/35. Also, the asymptotic density of numbers divisible by the first n primes is r(n) where r(1) = 1/2 and r(n) = r(n - 1) + (1 - r(n - 1)) / prime(n). - David A. Corneth, May 28 2017

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A187398, A187516, A187238, A187533, A187534, A187551.

digDivQ[n_] := AnyTrue[IntegerDigits[n], # > 1 && Mod[n, #] == 0 &]; Select[Range[200], digDivQ] (* Giovanni Resta, May 27 2017 *)

is(n) = my(d = vecsort(digits(n), , 8), t = 1); while(t<=#d&&d[t] < 2, t++); sum(i=t, #d, n%d[i]==0) > 0 \\ David A. Corneth, May 27 2017

Name edited by Alonso del Arte, May 16 2017

A235052 a(n) = smallest number > 1 such that a(n)^n contains a(n) as a substring.

Original entry on oeis.org

2, 5, 4, 5, 2, 4, 2, 2, 2, 2, 2, 3, 2, 4, 2, 3, 2, 2, 2, 3, 2, 3, 3, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 4, 3, 3, 2, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2

Offset: 1

Derek Orr, Jan 02 2014

It is very probable that a(n) = 2 for n > 169.

a(n) <= 5 since 5^n ends in 5 for n > 0. - Michael S. Branicky, Jan 25 2022

			4 is the smallest number such that 4^3 contains a 4 and 4^6 contains a 4, so a(3) = 4 and a(6) = 4.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A034293, A236046.

import Data.List (isInfixOf)
a235052 n = head [x | x <- [2..], show x `isInfixOf` (show $ x ^ n)]
-- Reinhard Zumkeller, Jan 18 2014

a[n_] := Block[{k = 2}, While[StringPosition[ToString[k^n], ToString@k] == {}, k++]; k]; Array[a, 80] (* Giovanni Resta, Jan 11 2014 *)

a(n) = my(k=2); while(#strsplit(Str(k^n), Str(k))==1, k++); k; \\ Michel Marcus, Jan 25 2022

def f(x):
  for n in range(2,10**3):
    if str(n**x).count(str(n)) > 0:
      return n
x = 1
while x < 200:
  print(f(x))
  x += 1

def a(n): return min(k for k in range(2, 6) if str(k) in str(k**n))
print([a(n) for n in range(1, 88)]) # Michael S. Branicky, Jan 25 2022

a(81) and beyond from Michael S. Branicky, Jan 25 2022

A236046 Smallest number m > 1 such that m^n does not contain m as a substring.

Original entry on oeis.org

2, 2, 2, 11, 2, 3, 3, 11, 3, 3, 2, 11, 2, 11, 2, 11, 11, 3, 2, 14, 2, 2, 7, 11, 2, 3, 3, 11, 11, 12, 14, 11, 2, 2, 2, 11, 11, 2, 3, 14, 2, 13, 12, 11, 2, 11, 12, 11, 3, 14, 11, 11, 2, 3, 11, 11, 12, 11, 11, 14, 12, 11, 2, 11, 12, 11, 11, 13, 11, 13, 13, 18

Offset: 2

Reinhard Zumkeller, Jan 18 2014

Reinhard Zumkeller, Table of n, a(n) for n = 2..5000

Cf. A034293, A235052.

import Data.List (isInfixOf)
a236046 n = head [x | x <- [2..], not $ show x `isInfixOf` (show $ x ^ n)]

a(n) = my(k=2); while(#strsplit(Str(k^n), Str(k))!=1, k++); k; \\ Michel Marcus, Jan 25 2022

A378556 Powers of 2 that do not include the digit 2.

Original entry on oeis.org

1, 4, 8, 16, 64, 4096, 16384, 65536, 1048576, 4194304, 8388608, 67108864, 17179869184, 34359738368, 68719476736, 549755813888, 4398046511104, 70368744177664, 18014398509481984, 18446744073709551616, 18889465931478580854784, 9671406556917033397649408, 374144419156711147060143317175368453031918731001856

Offset: 1

Erich Friedman, Nov 30 2024

Any additional terms have exponent at least 10^5.

Cf. A000079, A034293.

Select[2^Range[0,100000],Not[MemberQ[IntegerDigits[#],2]]&]

a(n) = 2^A034293(n).

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

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

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A035064 Numbers k such that 2^k does not contain the digit 9 (probably finite).

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A113951 Largest number whose n-th power is exclusionary (or 0 if no such number exists).

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A136291 Array read by rows: each row is a sequence of numbers k such that n^k does not contain the digit n.

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A185186 Numbers divisible by at least one of their digits other than 1.

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