A052382 -id:A052382 - cp's OEIS frontend

A055640 Number of nonzero digits in decimal expansion of n.

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2

Offset: 0

Henry Bottomley, Jun 06 2000

Comment from Antti Karttunen, Sep 05 2004: (Start)
Also number of characters needed to write the number n in classical Greek alphabetic system, up to n=999. The Greek alphabetic system assigned values to the letters as follows:
alpha = 1, beta = 2, gamma = 3, delta = 4, epsilon = 5, digamma = 6, zeta = 7, eta = 8, theta = 9, iota = 10, kappa = 20, lambda = 30, mu = 40, nu = 50, xi = 60, omicron = 70, pi = 80, koppa = 90, rho = 100, sigma = 200, tau = 300, upsilon = 400, phi = 500, chi = 600, psi = 700, omega = 800, sampi = 900. (End)
For partial sums see A102685. - Hieronymus Fischer, Jun 06 2012

			129 is written as rho kappa theta in the old Greek system.

L. Threatte, The Greek Alphabet, in The World's Writing Systems, edited by Peter T. Daniels and William Bright, Oxford Univ. Press, 1996, p. 278.

Hieronymus Fischer, Table of n, a(n) for n = 0..10000
Unicode Consortium, Unicode Home Page. (Follow the link "Display Problems?" to find an appropriate information/font file to show the Greek characters correctly. Or look at the HTML source to see their names.)

Cf. A102669-A102685, A004719, A011540, A052382, A061745, A054899, A055641, A055642, A122840, A160093, A160094, A193238, A196563, A195564, A000120, A000788, A023416, A059015 (for base 2).

Differs from A098378 for the first time at position n=200 with a(200)=1, as only one nonzero Arabic digit (and only one Greek letter) is needed for two hundred, while A098378(200)=2 as two characters are needed in the Ethiopic system.

a055640 n = length $ filter (/= '0') $ show n
-- Reinhard Zumkeller, May 02 2011

Table[Count[IntegerDigits[n],?(#>0&)],{n,0,120}] (* _Harvey P. Dale, Mar 11 2012 *)
Total[Most[DigitCount[#]]]&/@Range[0,120] (* Harvey P. Dale, Mar 19 2021 *)

a(n)=my(v=digits(n));sum(i=1,#v,!!v[i]) \\ Charles R Greathouse IV, Aug 05 2012

From Hieronymus Fischer, Jun 06 2012: (Start)
a(n) = Sum_{j=1..m+1} (floor(n/10^j+0.9) - floor(n/10^j)), where m = floor(log_10(n)).
a(n) = m + 1 - A055641(n).
G.f.: (1/(1-x))*Sum_{j>=0} (x^10^j - x^(10*10^j))/(1-x^10^(j+1)). (End)
a(n) = A055642(n) - A055641(n).

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

Original entry on oeis.org

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)

A034838 Numbers k that are divisible by every digit of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 22, 24, 33, 36, 44, 48, 55, 66, 77, 88, 99, 111, 112, 115, 122, 124, 126, 128, 132, 135, 144, 155, 162, 168, 175, 184, 212, 216, 222, 224, 244, 248, 264, 288, 312, 315, 324, 333, 336, 366, 384, 396, 412, 424, 432, 444, 448

Offset: 1

Erich Friedman

Subset of zeroless numbers A052382: Integers with at least one digit 0 (A011540) are excluded.
A128635(a(n)) = n.
Contains in particular all repdigits A010785 \ {0}. - M. F. Hasler, Jan 05 2020
The greatest term such that the digits are all different is the greatest Lynch-Bell number 9867312 = A115569(548) = A113028(10) [see Diophante link]. - Bernard Schott, Mar 18 2021
Named "nude numbers" by Katagiri (1982-83). - Amiram Eldar, Jun 26 2021

			36 is in the sequence because it is divisible by both 3 and 6.
48 is included because both 4 and 8 divide 48.
64 is not included because even though 4 divides 64, 6 does not.

Charles Ashbacher, Journal of Recreational Mathematics, Vol. 33 (2005), pp. 227. See problem number 2693.
Yoshinao Katagiri, Letter to the editor of the Journal of Recreational Mathematics, Vol. 15, No. 4 (1982-83).
Margaret J. Kenney and Stanley J. Bezuszka, Number Treasury 3: Investigations, Facts And Conjectures About More Than 100 Number Families, World Scientific, 2015, p. 175.
Thomas Koshy, Elementary Number Theory with Applications, Elsevier, 2007, p. 79.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Diophante, A1916. Le plus grand entier divisible par ses propres chiffres (in French).
Giovanni Resta, nude numbers, Numbersaplenty, 2013.
Roberto A. Ribas, The Nude Numbers, The Pentagon, Vol. 45, No. 1 (1985), pp. 18-31.
Voodooguru, Nude Numbers, Mathematical Meanderings, Oct 11 2020.
Eric Weisstein's World of Mathematics, Digit
Index entries for 10-automatic sequences.

Intersection of A002796 (numbers divisible by each nonzero digit) and A052382 (zeroless numbers), or A002796 \ A011540 (numbers with digit 0).
Subsequence of A034709 (divisible by last digit).
Contains A007602 (multiples of the product of their digits) and subset A059405 (n is the product of its digits raised to positive powers), A225299 (divisible by square of each digit), and A066484 (n and its rotations are divisible by each digit).
Cf. A113028, A346267 (number of terms with n digits), A087140 (complement).
Supersequence of A115569 (with all different digits).

a034838 n = a034838_list !! (n-1)
a034838_list = filter f a052382_list where
   f u = g u where
     g v = v == 0 || mod u d == 0 && g v' where (v',d) = divMod v 10
-- Reinhard Zumkeller, Jun 15 2012, Dec 21 2011

[n:n in [1..500]| not 0 in Intseq(n) and #[c:c in [1..#Intseq(n)]| n mod Intseq(n)[c] eq 0] eq #Intseq(n)] // Marius A. Burtea, Sep 12 2019

a:=proc(n) local nn,j,b,bb: nn:=convert(n,base,10): for j from 1 to nops(nn) do b[j]:=n/nn[j] od: bb:=[seq(b[j],j=1..nops(nn))]: if map(floor,bb)=bb then n else fi end: 1,2,3,4,5,6,7,8,9,seq(seq(seq(a(100*m+10*n+k),k=1..9),n=1..9),m=0..6); # Emeric Deutsch

divByEvryDigitQ[n_] := Block[{id = Union[IntegerDigits[n]]}, Union[ IntegerQ[ #] & /@ (n/id)] == {True}]; Select[ Range[ 487],  divByEvryDigitQ[#] &] (* Robert G. Wilson v, Jun 21 2005 *)
Select[Range[500],FreeQ[IntegerDigits[#],0]&&AllTrue[#/ IntegerDigits[ #], IntegerQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 31 2019 *)

is(n)=my(v=vecsort(eval(Vec(Str(n))),,8)); if(v[1]==0, return(0)); for(i=1, #v, if(n%v[i], return(0))); 1 \\ Charles R Greathouse IV, Apr 17 2012

is_A034838(n)=my(d=Set(digits(n)));d[1]&&!forstep(i=#d,1,-1,n%d[i]&&return) \\ M. F. Hasler, Jan 10 2016

A034838_list = []
for g in range(1,4):
    for n in product('123456789',repeat=g):
        s = ''.join(n)
        m = int(s)
        if not any(m % int(d) for d in s):
            A034838_list.append(m) # Chai Wah Wu, Sep 18 2014

for n in range(10**3):
    s = str(n)
    if '0' not in s:
        c = 0
        for i in s:
            if n%int(i):
                c += 1
                break
        if not c:
            print(n,end=', ') # Derek Orr, Sep 19 2014

# finite automaton accepting sequence (see comments in A346267)
from math import gcd
def lcm(a, b): return a * b // gcd(a, b)
def inF(q): return q[0]%q[1] == 0
def delta(q, c): return ((10*q[0]+c)%2520, lcm(q[1], c))
def ok(n):
    q = (0, 1)
    for c in map(int, str(n)):
        if c == 0: return False # computation dies
        else: q = delta(q, c)
    return inF(q)
print(list(filter(ok, range(450)))) # Michael S. Branicky, Jul 18 2021

A067251 Numbers with no trailing zeros in decimal representation.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101, 102, 103, 104

Offset: 1

Reinhard Zumkeller, Mar 10 2002

Or, decimated numbers: every 10th number has been omitted from the natural numbers. - Cino Hilliard, Feb 21 2005. For example, The 10th number starting with 1 is 10 and is missing from the table because it was decimated.
The word "decimated" can be interpreted in several ways and should be used with caution. - N. J. A. Sloane, Feb 21 2005
Not the same as A052382, as 101 is included.
Numbers in here but not in A043095 are 81, 91, 92, 93, 94,... for example. - R. J. Mathar, Sep 30 2008
The integers 100*a(n) are precisely the numbers whose square ends with exactly 4 identical digits while the integers 10*a(n) form just a subsequence of the numbers whose square ends with exactly 2 identical digits (A346678). - Bernard Schott, Oct 04 2021

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,1,-1).

Complement of A008592.
Cf. A076641 (reversed).
Cf. A004086, A043095, A052382, A168184.
Cf. A039685 (a subsequence), A346678, A346940, A346942.

a067251 n = a067251_list !! (n-1)
a067251_list = filter ((> 0) . flip mod 10) [0..]
-- Reinhard Zumkeller, Jul 11 2015, Dec 29 2011

S := seq(n + floor((n-1)/9), n=1..100); # Bernard Schott, Oct 04 2021

DeleteCases[Range[110],?(Divisible[#,10]&)] (* _Harvey P. Dale, May 16 2016 *)

f(n) = for(x=1,n,if(x%10,print1(x","))) \\ Cino Hilliard, Feb 21 2005

Vec(x*(x+1)*(x^4-x^3+x^2-x+1)*(x^4+x^3+x^2+x+1)/((x-1)^2*(x^2+x+1)*(x^6+x^3+1)) + O(x^100)) \\ Colin Barker, Sep 28 2015

def a(n): return n + (n-1)//9
print([a(n) for n in range(1, 95)]) # Michael S. Branicky, Oct 04 2021

a(n) = n + floor((n-1)/9).
a(n) mod 10 > 0 for all n.
A004086(A004086(a(n))) = a(n).
A168184(a(n)) = 1. - Reinhard Zumkeller, Nov 30 2009
From Colin Barker, Sep 28 2015: (Start)
a(n) = a(n-1) + a(n-9) - a(n-10) for n>10.
G.f.: x*(x+1)*(x^4-x^3+x^2-x+1)*(x^4+x^3+x^2+x+1) / ((x-1)^2*(x^2+x+1)*(x^6+x^3+1)). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (1/20 + 1/sqrt(5) - sqrt(1+2/sqrt(5))/5) * Pi. - Amiram Eldar, May 11 2025

Edited by N. J. A. Sloane, Sep 06 2008 at the suggestion of R. J. Mathar

Typos corrected in a comment line by Reinhard Zumkeller, Apr 04 2010

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.

A052383 Numbers without 1 as a digit.

Original entry on oeis.org

0, 2, 3, 4, 5, 6, 7, 8, 9, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89

Offset: 1

Henry Bottomley, Mar 13 2000

For each k in {1, 2, ..., 29, 30, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43} there exists at least an m such that m^k is 1-less. If m^k is 1-less then (10*m)^k, (100*m)^k, (1000*m)^k, ... are also 1-less. Therefore for each of these numbers k there exist infinitely many k-th powers in this sequence. - Mohammed Yaseen, Apr 17 2023

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
M. F. Hasler, Numbers avoiding certain digits, OEIS Wiki, Jan 12 2020.
Index entries for 10-automatic sequences.

Cf. A004176, A004720, A011531 (complement), A038603 (subset of primes), A082830 (Kempner series), A248518, A248519.

Cf. A052382 (without 0), A052404 (without 2), A052405 (without 3), A052406 (without 4), A052413 (without 5), A052414 (without 6), A052419 (without 7), A052421 (without 8), A007095 (without 9).

a052383 = f . subtract 1 where
   f 0 = 0
   f v = 10 * f w + if r > 0 then r + 1 else 0  where (w, r) = divMod v 9
-- Reinhard Zumkeller, Oct 07 2014

[ n: n in [0..89] | not 1 in Intseq(n) ];  // Bruno Berselli, May 28 2011

M:= 3: # to get all terms with up to M digits
B:= {$2..9}: A:= B union {0}:
for m from 1 to M do
B:= map(b -> seq(10*b+j,j={0,$2..9}), B);
A:= A union B;
od:
sort(convert(A,list)); # Robert Israel, Jan 11 2016
# second program:
A052383 := proc(n)
      option remember;
      if n = 1 then
        0;
    else
        for a from procname(n-1)+1 do
            if nops(convert(convert(a,base,10),set) intersect {1}) = 0 then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Jul 31 2016
# third Maple program:
a:= proc(n) local l, m; l, m:= 0, n-1;
      while m>0 do l:= (d->
        `if`(d=0, 0, d+1))(irem(m, 9, 'm')), l
      od; parse(cat(l))/10
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 01 2016

ban1Q[n_] := FreeQ[IntegerDigits[n], 1] == True; Select[Range[0, 89], ban1Q[#] &] (* Jayanta Basu, May 17 2013 *)
Select[Range[0, 99], DigitCount[#, 10, 1] == 0 &] (* Alonso del Arte, Jan 12 2020 *)

a(n)=my(v=digits(n,9));for(i=1,#v,if(v[i],v[i]++));subst(Pol(v),'x,10) \\ Charles R Greathouse IV, Oct 04 2012

apply( {A052383(n)=fromdigits(apply(d->d+!!d, digits(n-1, 9)))}, [1..99]) \\ Defines the function and computes it for indices 1..99 (check & illustration)
select( {is_A052383(n)=!setsearch(Set(digits(n)), 1)}, [0..99]) \\ Define the characteristic function is_A; as illustration, select the terms in [0..99]
next_A052383(n, d=digits(n+=1))={for(i=1, #d, d[i]==1&& return((1+n\d=10^(#d-i))*d)); n} \\ Successor function: efficiently skip to the next a(k) > n. Used in A038603.  - M. F. Hasler, Jan 11 2020

from itertools import count, islice, product
def A052383(): # generator of terms
    yield 0
    for digits in count(1):
        for f in "23456789":
            for r in product("023456789", repeat=digits-1):
                yield int(f+"".join(r))
print(list(islice(A052383(), 72))) # Michael S. Branicky, Oct 15 2023

from gmpy2 import digits
def A052383(n): return int(''.join(str(int(d)+(d!='0')) for d in digits(n-1,9))) # Chai Wah Wu, Jun 28 2025

(0 to 99).filter(.toString.indexOf('1') == -1) // _Alonso del Arte, Jan 12 2020

seq 0 1000 | grep -v 1; # Joerg Arndt, May 29 2011

a(1) = 1, a(n + 1) = f(a(n) + 1, a(n) + 1) where f(x, y) = if x < 10 and x <> 1 then y else if x mod 10 = 1 then f(y + 1, y + 1) else f(floor(x/10), y). - Reinhard Zumkeller, Mar 02 2008
a(n) is the replacement of all nonzero digits d by d + 1 in the base-9 representation of n - 1. - Reinhard Zumkeller, Oct 07 2014
Sum_{k>1} 1/a(k) = A082830 = 16.176969... (Kempner series). - Bernard Schott, Jan 12 2020

Offset changed by Reinhard Zumkeller, Oct 07 2014

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

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

A055640 Number of nonzero digits in decimal expansion of n.

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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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A034838 Numbers k that are divisible by every digit of k.

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A067251 Numbers with no trailing zeros in decimal representation.

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

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

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