A007377 -id:A007377 - cp's OEIS frontend

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

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

A238939 Powers of 3 without the digit '0' in their decimal expansion.

Original entry on oeis.org

1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 177147, 531441, 1594323, 4782969, 1162261467, 94143178827, 282429536481, 2541865828329, 7625597484987, 22876792454961, 617673396283947, 16677181699666569, 278128389443693511257285776231761

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, 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.

Select[3^Range[0,100],DigitCount[#,10,0]==0&] (* Paolo Xausa, Oct 07 2023 *)

for(n=0,99,vecmin(digits(3^n))&& print1(3^n","))

a(n) = 3^A030700(n).

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

A103663 Smallest integer base x > 1 such that x^n has no digit 0 in its decimal representation, or 0 if no such x exists.

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 3, 3, 2, 2, 2, 2, 4, 2, 2, 12, 381, 22, 3, 2, 2, 3, 2, 2, 6, 5, 2, 2, 2, 2, 2, 2, 2, 4, 2, 0, 11, 0, 4, 6, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 0, 2, 0, 0, 0, 2, 2, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Hugo Pfoertner, Feb 28 2005

Bases associated with A103662. [Corrected by M. F. Hasler, Mar 09 2014]

Zero values are conjectural. a(40) was checked up to 10^9 by Joshua Zucker. All other zero values were checked up to 10^5 by David Wasserman.

			a(10) = 5 because 5^10 is the smallest 10th power containing no zero in its decimal representation (2^10 = 1024, 3^10 = 59049, 4^10 = 1048576).

Cf. A103662 = smallest power x^n with integer x>1 that has no digit 0 in its decimal representation.

A007377 gives n such that a(n) = 2. Cf. A008839, A030700-A030706.

a(21) = 381 found by Joshua Zucker

More terms from David Wasserman, Apr 17 2008

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

Original entry on oeis.org

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.

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

A130694 Exponents of powers of 2 that contain all ten digits.

Original entry on oeis.org

68, 70, 79, 82, 84, 87, 88, 89, 94, 95, 96, 97, 98, 100, 101, 103, 104, 105, 106, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144

Offset: 1

Greg Dresden, Jul 10 2007

It is believed that every power of 2 beyond 2^86 contains the digit 0.

For k in {51,67,72,76,81,86}, 2^k contains all nonzero digits, but does not contain 0. - Dimiter Skordev, Oct 05 2021

			2^68 = 295147905179352825856.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A007377, A000079.

Complement of A130696.

A2 := {}; Do[If[Length[Union[ IntegerDigits[2^ n]]] == 10, A2 = Join[A2, {n}]], {n, 1, 200}]; Print[A2]
Select[Range[200],Min[DigitCount[2^#]]>0&] (* Harvey P. Dale, Aug 03 2019 *)

is_A130694(n)=9<#Set(Vec(Str(1<M. F. Hasler, Aug 25 2012

A043537(A000079(a(n))) = 10. - Reinhard Zumkeller, Jul 29 2007

a(n) = n + 91 for n >= 78 (conjectured). - Chai Wah Wu, Jan 27 2020

Displayed terms double-checked by M. F. Hasler, Aug 25 2012

A305932 Irregular table: row n >= 0 lists all k >= 0 such that the decimal representation of 2^k has n digits '0' (conjectured).

Original entry on oeis.org

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, 10, 11, 12, 17, 20, 21, 22, 23, 26, 29, 30, 38, 40, 41, 44, 45, 46, 47, 48, 50, 57, 58, 65, 66, 68, 71, 73, 74, 75, 84, 85, 95, 96, 122, 124, 129, 130, 149, 151, 184, 43, 53, 61, 69, 70

Offset: 0

M. F. Hasler, Jun 14 2018

A partition of the nonnegative integers (the rows being the subsets).
Although it remains an open problem to provide a proof that the rows are complete (as are all terms of A020665), we can assume it for the purpose of this sequence.
Read as a flattened sequence, a permutation of the nonnegative integers.

			The table reads:
n \ k's
0 : 0, 1, ..., 9, 13, 14, 15, 16, 18, 19, 24, 25, 27, (...), 81, 86 (cf. A007377)
1 : 10, 11, 12, 17, 20, 21, 22, 23, 26, 29, 30, 38, 40, 41, 44, (...), 151, 184
2 : 42, 52, 54, 55, 56, 59, 60, 62, 63, 64, 78, 80, 82, 92, 107, (...), 171, 231
3 : 43, 53, 61, 69, 70, 83, 87, 89, 90, 93, 109, 112, 114, 115, (...), 221, 359
4 : 79, 91, 94, 97, 106, 118, 126, 127, 137, 139, 157, 159, 170, (...), 241, 283
5 : 88, 98, 99, 103, 104, 113, 120, 143, 144, 146, 152, 158, 160, (...), 343, 357
...
Column 0 is A031146: least k such that 2^k has n digits '0' in base 10.
Row lengths = number of powers of 2 with exactly n '0's = (36, 41, 31, 34, 25, 32, 37, 23, 43, 47, 33, 35, 29, 27, 27, 39, 34, 34, 28, 29, ...): not in the OEIS.
Largest number in row n = (86, 229, 231, 359, 283, 357, 475, 476, 649, 733, 648, 696, 824, 634, 732, 890, 895, 848, 823, 929, 1092, ...): not in the OEIS.
Row number of n = Number of '0's in 2^n = A027870: (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, ...).
Inverse permutation (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 36, 37, 38, 10, 11, 12, 13, 39, 14, 15, 40, 41, 42, 43, 16, 17, 44, 18, 19, 45, 46, 20, 21, ...) is not in the OEIS.

M. F. Hasler, Zeroless powers, OEIS Wiki, March 2014.

Cf. A007377, A031146.
Sequence A027870 yields the row number of a given integer.
Cf. A305933 (analog for 3^n), A305924 (for 4^n), ..., A305929 (for 9^n).

mx = 1000; g[n_] := g[n] = DigitCount[2^n, 10, 0]; f[n_] := Select[Range@mx, g@# == n &]; Table[f@n, {n, 0, 4}] // Flatten (* Robert G. Wilson v, Jun 20 2018 *)

apply( A305932_row(n,M=200*(n+1))=select(k->A027870(k)==n,[0..M]), [0..20]) \\ A027870(k)=#select(d->!d, digits(2^k))

Row n = { k >= 0 | A027870(k) = n }.

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

A071531 Smallest exponent r such that n^r contains at least one zero digit (in base 10).

Original entry on oeis.org

10, 10, 5, 8, 9, 4, 4, 5, 1, 5, 4, 6, 7, 4, 3, 7, 4, 4, 1, 5, 3, 6, 6, 4, 6, 5, 5, 4, 1, 6, 2, 2, 3, 4, 5, 3, 4, 5, 1, 5, 3, 3, 4, 2, 5, 2, 2, 2, 1, 2, 2, 2, 4, 2, 5, 4, 6, 3, 1, 5, 6, 3, 2, 4, 6, 3, 9, 3, 1, 2, 6, 3, 3, 4, 8, 4, 2, 3, 1, 4, 5, 5, 2, 4, 3, 3, 6, 3, 1, 5, 5, 3, 3, 2, 7, 2, 2, 2, 1, 1, 1, 1, 1, 1

Offset: 2

Paul Stoeber (paul.stoeber(AT)stud.tu-ilmenau.de), Jun 02 2002

For all n, a(n) is at most 40000, as shown below. Is 10 an upper bound?
If n has d digits, the numbers n, n^2, ..., n^k have a total of about N = k*(k+1)*d/2, and if these were chosen randomly the probability of having no zeros would be (9/10)^N. The expected number of d-digit numbers n with f(n)>k would be 9*10^(d-1)*(9/10)^N. If k >= 7, (9/10)^(k*(k+1)/2)*10 < 1 so we would expect heuristically that there should be only finitely many n with f(n) > 7. - Robert Israel, Jan 15 2015
The similar definition using "...exactly one digit 0..." would be ill-defined for all multiples of 100 and others (1001, ...). - M. F. Hasler, Jun 25 2018
When r=40000, one of the last five digits of n^r is always 0. Working modulo 10^5, we have 2^r=9736 and 5^r=90625, and both of these are idempotent; also, if gcd(n,10)=1, then n^r=1, and if 10|n, then n^r=0. Therefore the last five digits of n^r are always either 00000, 00001, 09736, or 90625. In particular, a(n) <= 40000. - Mikhail Lavrov, Nov 18 2021

			a(4)=5 because 4^1=4, 4^2=16, 4^3=64, 4^4=256, 4^5=1024 (has zero digit).

Robert Israel, Table of n, a(n) for n = 2..10000
OEIS Wiki, Zeroless powers (2014).

Cf. A305941 for the actual powers n^k.
Cf. A007377, A030700, A030701, A008839, A030702, A030703, A030704, A030705, A030706, A195944: decimal expansion of k^n contains no zeros, k = 2, 3, 4, ...
Cf. A305932, A305933, A305924, ..., A305929: row n = {k: x^k has n 0's}, x = 2, 3, ..., 9.
Cf. A305942, ..., A305947, A305938, A305939: #{k: x^k has n 0's}, x = 2, 3, ..., 9.
Cf. A306112, ..., A306119: largest k: x^k has n 0's; x = 2, 3, ..., 9.

f:= proc(n) local j;
for j from 1 do if has(convert(n^j,base,10),0) then return j fi od:
end proc:
seq(f(n),n=2..100); # Robert Israel, Jan 15 2015

zd[n_]:=Module[{r=1},While[DigitCount[n^r,10,0]==0,r++];r]; Array[zd,110,2] (* Harvey P. Dale, Apr 15 2012 *)

A071531(n)=for(k=1, oo, vecmin(digits(n^k))||return(k)) \\ M. F. Hasler, Jun 23 2018

def a(n):
    r, p = 1, n
    while 1:
        if "0" in str(p):
            return r
        r += 1
        p *= n
[a(n) for n in range(2, 100)] # Tim Peters, May 19 2005

a(n) >= 1 with equality iff n is in A011540 \ {0} = {10, 20, ..., 100, 101, ...}. - M. F. Hasler, Jun 23 2018

cp's OEIS Frontend

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

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A238939 Powers of 3 without the digit '0' in their decimal expansion.

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A103663 Smallest integer base x > 1 such that x^n has no digit 0 in its decimal representation, or 0 if no such x exists.

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

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A130694 Exponents of powers of 2 that contain all ten digits.

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