A007377 -id:A007377 - 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

A027870 Number of zero digits in 2^n.

Original entry on oeis.org

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, 0, 0, 0, 1, 0, 1, 1, 2, 3, 1, 1, 1, 1, 1, 0, 1, 0, 2, 3, 2, 2, 2, 1, 1, 2, 2, 3, 2, 2, 2, 1, 1, 0, 1, 3, 3, 1, 0, 1, 1, 1, 0, 0, 2, 4, 2, 0, 2, 3, 1, 1, 0, 3, 5, 3, 3, 4, 2, 3, 4, 1, 1, 4, 5, 5, 6, 6, 7, 5, 5

Offset: 0

N. J. A. Sloane

I conjecture that any value x = 0, 1, 2, ... occurs only a finite number of times N(x) = 36, 41, 31, 34, 25, 32, 37, 23, 43, 47, 33, ... in this sequence, for the last time at well defined indices i(x) = 86, 229, 231, 359, 283, 357, 475, 476, 649, 733, 648, ... - M. F. Hasler, Jul 09 2025

			2^31 = 2147483648 so a(31) = 0 and 2^42 = 4398046511104 so a(42) = 2.

Alois P. Heinz, Table of n, a(n) for n = 0..20000 (first 1001 terms from _Harry J. Smith_, Oct 27 2009)

Cf. A000079 (powers of 2), A007377 (2^n has no zeros).
Similar for other digits: A065712 (1's), A065710 (2's), A065714 (3's), A065715 (4's), A065716 (5's), A065717 (6's), A065718 (7's), A065719 (8's), A065744 (9's).
Cf. A031146 (index of first appearance of n in this sequence), A094776 (index of last occurrence of digit n in powers of 2).
Cf. A305932 (table with n in row a(n)).

a027870 = a055641 . a000079  -- Reinhard Zumkeller, Apr 30 2013

Table[ Count[ IntegerDigits[2^n], 0], {n, 0, 100} ]
DigitCount[2^Range[0,110],10,0] (* Harvey P. Dale, Nov 20 2011 *)

A027870(n)=#select(d->!d,digits(2^n)) \\ M. F. Hasler, Jun 14 2018

def A027870(n):
    return str(2**n).count('0') # Chai Wah Wu, Feb 14 2020

a(n) = A055641(A000079(n)). - Reinhard Zumkeller, Apr 30 2013

a(A007377(n)) = 0; A224782(n) <= a(n). - Reinhard Zumkeller, Apr 30 2013

Edited by M. F. Hasler, Jul 09 2025

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.

A007496 Numbers n such that the decimal expansions of 2^n and 5^n contain no 0's (probably 33 is last term).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 18, 33

Offset: 1

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

Intersection of A007377 and A008839. - Lekraj Beedassy, Jul 27 2004
From Jonathan Vos Post, Jul 20 2005: (Start)
Equivalently, numbers n such that 10^n is the product of two integers without any zero digits.
  10^0 = 1 * 1
  10^1 = 2 * 5
  10^2 = 4 * 25
  10^3 = 8 * 125
  10^4 = 16 * 625
  10^5 = 32 * 3125
  10^6 = 64 * 15625
  10^7 = 128 * 78125
  10^9 = 512 * 1953125
  10^18 = 262144 * 3814697265625
  10^33 = 8589934592 * 116415321826934814453125. (End)
Searched for n up to 10^10. - David Radcliffe, Dec 27 2015

J. S. Madachy, Madachy's Mathematical Recreation, "#2. Number Toughies", pp. 126-8, Dover NY 1979.
C. S. Ogilvy and J. T. Anderson, Excursions in Number Theory. Oxford Univ. Press, 1966, p. 89.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Leroy C. Dalton & Henry D. Snyder, Topics for Mathematics Clubs, pp. 68-69, NCTM Reston VA 1983.
G. P. Michon, What two integers without zero digits have a product of 1000000000?
C. S. Ogilvy and J. T. Anderson, Excursions in Number Theory, Oxford Univ. Press, 1966, p. 89. (Annotated scanned copy).
W. Schneider, NoZeros: Powers n^k without Digit Zero [Cached copy]

Cf. A007377, A008839.

q:= n-> andmap(t-> not 0 in convert(t, base, 10), [2^n, 5^n]):
select(q, [$0..40])[];  # Alois P. Heinz, Feb 03 2022

Range@(10^5) // Select[Last@DigitCount@(5^#) == 0 &] // Select[Last@DigitCount@(2^#) == 0 &] (* Hans Rudolf Widmer, Feb 02 2022 *)

isok(n) = vecmin(digits(2^n)) && vecmin(digits(5^n)); \\ Michel Marcus, Dec 28 2015

Edited by N. J. A. Sloane, Oct 24 2009 at the suggestion of M. F. Hasler

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

A030702 Decimal expansion of 6^n contains no zeros (probably finite).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 12, 17, 24, 29, 44

Offset: 1

Eric W. Weisstein

W. Schneider, NoZeros: Powers n^k without Digit Zero [Cached copy]
Eric Weisstein's World of Mathematics, Zero

Cf. A007377, A008839, A030700, A305926.

[n: n in [0..500] | not 0 in Intseq(6^n)]; // Vincenzo Librandi, Mar 08 2014

Select[Range[50],FreeQ[IntegerDigits[6^#],0]&] (* Harvey P. Dale, Feb 26 2017 *)

for(n=0, 199, vecmin(digits(6^n))&& print1(n", ")) \\ M. F. Hasler, Mar 07 2014

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

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

A030703 Decimal expansion of 7^n contains no zeros (probably finite).

Original entry on oeis.org

0, 1, 2, 3, 6, 7, 10, 11, 19, 35

Offset: 1

Eric W. Weisstein

No additional terms up to 20000. - Harvey P. Dale, Oct 02 2013

W. Schneider, NoZeros: Powers n^k without Digit Zero [Cached copy]
Eric Weisstein's World of Mathematics, Zero

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

[n: n in [0..500] | not 0 in Intseq(7^n)]; // Vincenzo Librandi, Mar 08 2014

Select[Range[0,100],DigitCount[7^#,10,0]==0&] (* Harvey P. Dale, Oct 02 2013 *)

for( n=0, 9999, is_A052382(7^n) && print1(n, ", "))  \\ M. F. Hasler, Sep 25 2011

A030703 = A000420^(-1)(A052382) as a set, where f^(-1)(Y) = { x : f(x) in Y}.

A030703 = A000420^(-1) o A195908 as a function. - M. F. Hasler, Sep 25 2011

Initial term 0 inserted by M. F. Hasler, Sep 25 2011

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

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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Maple

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

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Magma

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A027870 Number of zero digits in 2^n.

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

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A007496 Numbers n such that the decimal expansions of 2^n and 5^n contain no 0's (probably 33 is last term).

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Maple

Mathematica

PARI

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

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Magma

Mathematica

PARI