A238938 -id:A238938 - cp's OEIS frontend

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

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

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

A195946 Powers of 11 which have no zero in their decimal expansion.

Original entry on oeis.org

1, 11, 121, 1331, 14641, 1771561, 19487171, 214358881, 2357947691, 3138428376721, 34522712143931, 379749833583241, 4177248169415651, 45949729863572161, 5559917313492231481, 4978518112499354698647829163838661251242411

Offset: 1

M. F. Hasler, Sep 25 2011

Probably finite. Is 4978518112499354698647829163838661251242411 the largest term?

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 the zeroless numbers (powers x^n), see A195942, A195943, A238938, A238939, A238940, A195948, A238936, A195908, A195945.

For the corresponding exponents, see A007377, A008839, A030700, A030701, A030702, A030703, A030704, A030705, A030706, A195944.

[11^n: n in [0..3*10^4] | not 0 in Intseq(11^n)]; // Bruno Berselli, Sep 26 2011

Select[11^Range[0,50],DigitCount[#,10,0]==0&] (* Harvey P. Dale, Jan 27 2014 *)

for( n=0,9999, is_A052382(11^n) && print1(11^n,","))

a(n) = 11^A030706(n).

A195946 = A001020 intersect A052382.

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

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).

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.

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.

A103662 Smallest power with base>1 and exponent n whose decimal representation doesn't contain the digit 0.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 9765625, 177147, 531441, 8192, 16384, 32768, 65536, 17179869184, 262144, 524288, 3833759992447475122176, 1582794342217312156827221746448942623537121214738891981, 341427877364219557396646723584, 94143178827, 16777216

Offset: 0

Hugo Pfoertner, Feb 28 2005

For statistical reasons it is conjectured that the sequence is finite.
a(40), if it exists, is not known.
From M. F. Hasler, Mar 08 2014 (Start):
Finite means here that for some n, no such power > 1 exists. This might be the case already for n=40. Alternatively, the sequence might be defined for all n by letting a(n)=1 if no larger power exists. Then a(n)=n^A103663(n) for all n.
Differs from A238938 from n=10 on. See the OEIS wiki page for further information. (End)

			base / exponent n / base^exponent
2 1 2
2 2 4
2 3 8
2 4 16
2 5 32
2 6 64
2 7 128
2 8 256
2 9 512
5 10 9765625
3 11 177147
3 12 531441
2 13 8192
2 14 16384
2 15 32768
2 16 65536
4 17 17179869184
2 18 262144
2 19 524288
12 20 3833759992447475122176
381 21 1582794342217312156827221746448942623537121214738891981
22 22 341427877364219557396646723584
3 23 94143178827
...
a(10)=9765625 because 5^10 is the smallest 10th power containing no zero in its decimal representation (2^10=1024, 3^10=59049, 4^10=1048576).

Michel Marcus, Table of n, a(n) for n = 0..39
M. F. Hasler, Zeroless powers, OEIS wiki, Mar 07 2014

Cf. A103663 = smallest bases x such that x^n has a zero-less decimal representation.

A103662(n)=for(b=2,9e9,vecmin(digits(b^n))&&return(b^n)) \\ M. F. Hasler, Mar 08 2014

a(n) = A103663(n)^n. - M. F. Hasler, Mar 08 2014

a(21)=381^21 found by Joshua Zucker.
Edited by Charles R Greathouse IV, Aug 03 2010
Keyword:fini removed by Jianing Song, Jan 28 2023 as finiteness is only conjectured.

cp's OEIS Frontend

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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Magma

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

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Magma

Mathematica

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

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Magma

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

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Mathematica

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A195946 Powers of 11 which have no zero in their decimal expansion.

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Programs

Magma

Mathematica

PARI

Formula

Extensions

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

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