A052383 -id:A052383 - cp's OEIS frontend

A248518 Number of partitions of n into parts > 0 without 1 as digit, cf. A052383.

1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 11, 13, 19, 21, 29, 34, 44, 51, 66, 75, 96, 110, 136, 157, 193, 220, 267, 307, 367, 421, 501, 571, 677, 772, 905, 1033, 1207, 1371, 1595, 1812, 2096, 2377, 2741, 3101, 3564, 4028, 4608, 5203, 5938, 6688, 7612, 8564, 9719, 10919

Offset: 0

Reinhard Zumkeller, Oct 07 2014

Note that the definition says "1 as a DIGIT", not "1 as a PART". - N. J. A. Sloane, Jun 28 2017

			The full list of partitions of 10 is as follows:
[[1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 2],
    [1, 1, 1, 1, 1, 1, 2, 2], [1, 1, 1, 1, 2, 2, 2], [1, 1, 2, 2, 2, 2],
    [2, 2, 2, 2, 2], [1, 1, 1, 1, 1, 1, 1, 3], [1, 1, 1, 1, 1, 2, 3],
    [1, 1, 1, 2, 2, 3], [1, 2, 2, 2, 3], [1, 1, 1, 1, 3, 3], [1, 1, 2, 3, 3],
    [2, 2, 3, 3], [1, 3, 3, 3], [1, 1, 1, 1, 1, 1, 4], [1, 1, 1, 1, 2, 4],
    [1, 1, 2, 2, 4], [2, 2, 2, 4], [1, 1, 1, 3, 4], [1, 2, 3, 4], [3, 3, 4],
    [1, 1, 4, 4], [2, 4, 4], [1, 1, 1, 1, 1, 5], [1, 1, 1, 2, 5], [1, 2, 2, 5],
    [1, 1, 3, 5], [2, 3, 5], [1, 4, 5], [5, 5], [1, 1, 1, 1, 6], [1, 1, 2, 6],
    [2, 2, 6], [1, 3, 6], [4, 6], [1, 1, 1, 7], [1, 2, 7], [3, 7], [1, 1, 8],
    [2, 8], [1, 9], [10]]
If we excluse those that have a 1 in one of the parts, 11 partitions are left:
[[2, 2, 2, 2, 2], [2, 2, 3, 3], [2, 2, 2, 4], [3, 3, 4], [2, 4, 4], [2, 3, 5], [5, 5], [2, 2, 6], [4, 6], [3, 7], [2, 8]].
So a(10) = 11. - _N. J. A. Sloane_, Jun 28 2017
a(11) = #{9+2, 8+3, 7+4, 7+2+2, 6+5, 6+3+2, 5+4+2, 5+3+3, 5+2+2+2, 4+4+3, 4+3+2+2, 3+3+3+2, 3+2+2+2+2} = 13;
a(12) = #{9+3, 8+4, 8+2+2, 7+5, 7+3+2, 6+6, 6+4+2, 6+3+3, 6+2+2+2, 5+5+2, 5+4+3, 5+3+2+2, 4+4+4, 4+4+2+2, 4+3+3+2, 4+2+2+2+2, 3+3+3+3, 3+3+2+2+2, 6x2} = 19.

Cf. A052383, A248519.

a248518 = p $ tail a052383_list where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m

Table[Length[Select[IntegerPartitions[n],!MemberQ[Flatten[ IntegerDigits/@#],1]&]],{n,0,60}] (* Harvey P. Dale, Jun 28 2017 *)

A248519 Number of partitions of n into distinct parts > 0 without 1 as digit, cf. A052383.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 3, 3, 5, 4, 6, 6, 7, 8, 9, 9, 10, 11, 10, 13, 11, 14, 13, 15, 15, 17, 19, 20, 25, 26, 31, 35, 41, 46, 55, 60, 70, 78, 87, 97, 106, 119, 127, 141, 150, 162, 175, 186, 201, 214, 229, 247, 264, 285, 308, 333, 363, 394, 431, 470, 513, 565

Offset: 0

Reinhard Zumkeller, Oct 07 2014

			a(10) = #{8+2, 7+3, 6+4, 5+3+2} = 4;
a(11) = #{9+2, 8+3, 7+4, 6+5, 6+3+2, 5+4+2} = 6;
a(12) = #{9+3, 8+4, 7+5, 7+3+2, 6+4+2, 5+4+3} = 6.

Cf. A052383, A248518.

a248519 = p $ tail a052383_list where
   p _      0 = 1
   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m

A007095 Numbers in base 9.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84

Offset: 0

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

Also numbers without 9 as a digit.

Complement of A011539: A102683(a(n)) = 0; A068505(a(n)) != a(n)). - Reinhard Zumkeller, Dec 29 2011

Julian Havil, Gamma, Exploring Euler's Constant, Princeton University Press, Princeton and Oxford, 2003, page 34.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Nathaniel Johnston, Table of n, a(n) for n = 0..10000
M. F. Hasler, Numbers avoiding certain digits, OEIS wiki, Jan 12 2020
Robert G. Wilson v, Letter to N. J. A. Sloane, Sep. 1992
Index entries for 10-automatic sequences.

Cf. A000042 (base 1), A007088 (base 2), A007089 (base 3), A007090 (base 4), A007091 (base 5), A007092 (base 6), A007093 (base 7), A007094 (base 8); A057104, A037479.
Cf. A052382 (without 0), A052383 (without 1), A052404 (without 2), A052405 (without 3), A052406 (without 4), A052413 (without 5), A052414 (without 6), A052419 (without 7), A052421 (without 8).
Cf. A031076, A031087.
Cf. A082838.

a007095 = f . subtract 1 where
   f 0 = 0
   f v = 10 * f w + r   where (w, r) = divMod v 9
-- Reinhard Zumkeller, Oct 07 2014, Dec 29 2011

[ n: n in [0..74] | not 9 in Intseq(n) ];  // Bruno Berselli, May 28 2011

A007095 := proc(n) local l: if(n=0)then return 0: fi: l:=convert(n,base,9): return op(convert(l,base,10,10^nops(l))): end: seq(A007095(n),n=0..67); # Nathaniel Johnston, May 06 2011

Table[ FromDigits[ IntegerDigits[n, 9]], {n, 0, 75}]

a(n)=if(n<1,0,if(n%9,a(n-1)+1,10*a(n/9)))

A007095(n)=fromdigits(digits(n, 9)) \\ Michel Marcus, Dec 29 2018

# and others: see OEIS Wiki page (cf. LINKS).

from gmpy2 import digits
def A007095(n): return int(digits(n,9)) # Chai Wah Wu, May 06 2025

seq 0 1000 | grep -v 9; # Joerg Arndt, May 29 2011

a(0) = 0, a(n) = 10*a(n/9) if n==0 (mod 9), a(n) = a(n-1)+1 otherwise. - Benoit Cloitre, Dec 22 2002

Sum_{n>1} 1/a(n) = A082838 = 22.92067... (Kempner series). - Bernard Schott, Dec 29 2018; edited by M. F. Hasler, Jan 13 2020

A052382 Numbers without 0 in the decimal expansion, colloquial 'zeroless numbers'.

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, 111, 112, 113

Offset: 1

Henry Bottomley, Mar 13 2000

The entries 1 to 79 match the corresponding subsequence of A043095, but then 81, 91-98, 100, 102, etc. are only in one of the two sequences. - R. J. Mathar, Oct 13 2008
Complement of A011540; A168046(a(n)) = 1; A054054(a(n)) > 0; A007602, A038186, A038618, A052041, A052043, and A052045 are subsequences. - Reinhard Zumkeller, Apr 25 2012, Apr 07 2011, Dec 01 2009
a(n) = n written in base 9 where zeros are not allowed but nines are. The nine distinct digits used are 1, 2, 3, ..., 9 instead of 0, 1, 2, ..., 8. To obtain this sequence from the "canonical" base 9 sequence with zeros allowed, just replace any 0 with a 9 and then subtract one from the group of digits situated on the left. For example, 9^3 = 729 (10) (in base 10) = 1000 (9) (in base 9) = 889 (9-{0}) (in base 9 without zeros) because 100 (9) = [9-1]9 = 89 (9-{0}) and thus 1000 (9) = [89-1]9 = 889 (9-{0}). - Robin Garcia, Jan 15 2014
From Hieronymus Fischer, May 28 2014: (Start)
Inversion: Given a term m, the index n such that a(n) = m can be calculated by A052382_inverse(m) = m - sum_{1<=j<=k} floor(m/10^j)*9^(j-1), where k := floor(log_10(m)) [see Prog section for an implementation in Smalltalk].
Example 1: A052382_inverse(137) = 137 - (floor(137/10) + floor(137/100)*9) = 137 - (13*1 + 1*9) = 137 - 22 = 115.
Example 2: A052382_inverse(4321) = 4321 - (floor(4321/10) + floor(4321/100)*9 + floor(4321/1000)*81) = 4321 - (432*1 + 43*9 + 4*81) = 4321 - (432 + 387 + 324) = 3178. (End)
The sum of the reciprocals of these numbers from a(1)=1 to infinity, called the Kempner series, is convergent towards a limit: 23.103447... whose decimal expansion is in A082839. - Bernard Schott, Feb 23 2019
Integer n > 0 is encoded using bijective base-9 numeration, see Wikipedia link below. - Alois P. Heinz, Feb 16 2020

			For k >= 0, a(10^k) = (1, 11, 121, 1331, 14641, 162151, 1783661, 19731371, ...) = A325203(k). - _Hieronymus Fischer_, May 30 2012 and Jun 06 2012; edited by _M. F. Hasler_, Jan 13 2020

Paul Halmos, "Problems for Mathematicians, Young and Old", Dolciani Mathematical Expositions, 1991, p. 258.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
K. Mahler, On the generating function of the integers with a missing digit, J. Indian Math. Soc. 15A (1951), 34-40.
Eric Weisstein's World of Mathematics, Kempner Series.
Eric Weisstein's World of Mathematics, Zerofree
Wikipedia, Bijective numeration
Index entries for 10-automatic sequences.

Cf. A004719, A052040, different from A067251.
Cf. A055640, A046034, A007931, A007932, A084544, A084545.
Column k=9 of A214676.
Cf. A011540 (complement), A043489, A054054, A168046.
Cf. A052383 (without 1), A052404 (without 2), A052405 (without 3), A052406 (without 4), A052413 (without 5), A052414 (without 6), A052419 (without 7), A052421 (without 8), A007095 (without 9).
Zeroless numbers in some other bases <= 10: A000042 (base 2), A032924 (base 3), A023705 (base 4), A248910 (base 6), A255805 (base 8), A255808 (base 9).
Cf. A082839 (sum of reciprocals).
Cf. A038618 (subset of primes)
Subsequences: A007602, A038186, A052041, A052043, A052045, A059405, A066484, A099542.

a052382 n = a052382_list !! (n-1)
a052382_list = iterate f 1 where
f x = 1 + if r < 9 then x else 10 * f x' where (x', r) = divMod x 10
-- Reinhard Zumkeller, Mar 08 2015, Apr 07 2011

[ n: n in [1..114] | not 0 in Intseq(n) ]; // Bruno Berselli, May 28 2011

a:= proc(n) local d, l, m; m:= n; l:= NULL;
      while m>0 do d:= irem(m, 9, 'm');
        if d=0 then d:=9; m:= m-1 fi;
        l:= d, l
      od; parse(cat(l))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 11 2015
is_zeroless := n -> not is(0 in convert(n, base, 10)):
select(is_zeroless, [seq(1..113)]);  # Peter Luschny, Jun 20 2025

A052382 = Select[Range[100], DigitCount[#, 10, 0] == 0 &] (* Alonso del Arte, Mar 10 2011 *)

select( {is_A052382(n)=n&&vecmin(digits(n))}, [0..111]) \\ actually: is_A052382 = (bool) A054054. - M. F. Hasler, Jan 23 2013, edited Jan 13 2020

a(n) = for (w=0, oo, if (n >= 9^w, n -= 9^w, return ((10^w-1)/9 + fromdigits(digits(n, 9))))) \\ Rémy Sigrist, Jul 26 2017

apply( {A052382(n,L=logint(n,9))=fromdigits(digits(n-9^L>>3,9))+10^L\9}, [1..100])
next_A052382(n, d=digits(n+=1))={for(i=1, #d, d[i]|| return(n-n%(d=10^(#d-i+1))+d\9)); n} \\ least a(k) > n. Used in A038618.
( {A052382_vec(n,M=1)=M--;vector(n, i, M=next_A052382(M))} )(99) \\ n terms >= M
\\ See OEIS Wiki page (cf. LINKS) for more programs. - M. F. Hasler, Jan 11 2020

A052382 = [n for n in range(1,10**5) if not str(n).count('0')]
# Chai Wah Wu, Aug 26 2014

from sympy import integer_log
def A052382(n):
    m = integer_log(k:=(n<<3)+1,9)[0]
    return sum((1+(k-9**m)//(9**j<<3)%9)*10**j for j in range(m)) # Chai Wah Wu, Jun 27 2025

A052382
"Answers the n-th term of A052382, where n is the receiver."
^self zerofree: 10
A052382_inverse
"Answers that index n which satisfy A052382(n) = m, where m is the receiver.”
^self zerofree_inverse: 10
zerofree: base
"Answers the n-th zerofree number in base base, where n is the receiver. Valid for base > 2.
Usage: n zerofree: b [b = 10 for this sequence]
Answer: a(n)"
| n m s c bi ci d |
n := self.
c := base - 1.
m := (base - 2) * n + 1 integerFloorLog: c.
d := n - (((c raisedToInteger: m) - 1)//(base - 2)).
bi := 1.
ci := 1.
s := 0.
1 to: m
do:
[:i |
s := (d // ci \\ c + 1) * bi + s.
bi := base * bi.
ci := c * ci].
^s
zerofree_inverse: base
"Answers the index n such that the n-th zerofree number in base base is = m, where m is the receiver. Valid for base > 2.
Usage: m zerofree_inverse: b [b = 10 for this sequence]
Answer: n"
| m p q s |
m := self.
s := 0.
p := base.
q := 1.
[p < m] whileTrue:
[s := m // p * q + s.
p := base * p.
q := (base - 1) * q].
^m - s
"by Hieronymus Fischer, May 28 2014"

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

a(n+1) = f(a(n)) with f(x) = 1 + if x mod 10 < 9 then x else 10*f([x/10]). - Reinhard Zumkeller, Nov 15 2009
From Hieronymus Fischer, Apr 30, May 30, Jun 08 2012, Feb 17 2019: (Start)
a(n) = Sum_{j=0..m-1} (1 + b(j) mod 9)*10^j, where m = floor(log_9(8*n + 1)), b(j) = floor((8*n + 1 - 9^m)/(8*9^j)).
Also: a(n) = Sum_{j=0..m-1} (1 + A010878(b(j)))*10^j.
a(9*n + k) = 10*a(n) + k, k=1..9.
Special values:
a(k*(9^n - 1)/8) = k*(10^n - 1)/9, k=1..9.
a((17*9^n - 9)/8) = 2*10^n - 1.
a((9^n - 1)/8 - 1) = 10^(n-1) - 1, n > 1.
Inequalities:
a(n) <= (1/9)*((8*n+1)^(1/log_10(9)) - 1), equality holds for n=(9^k-1)/8, k>0.
a(n) > (1/10)*((8*n+1)^(1/log_10(9)) - 1), n > 0.
Lower and upper limits:
lim inf a(n)/10^log_9(8*n) = 1/10, for n -> infinity.
lim inf a(n)/n^(1/log_10(9)) = 8^(1/log_10(9))/10, for n -> infinity.
lim sup a(n)/10^log_9(8*n) = 1/9, for n -> infinity.
lim sup a(n)/n^(1/log_10(9)) = 8^(1/log_10(9))/9, for n -> infinity.
G.f.: g(x) = (x^(1/8)*(1-x))^(-1) Sum_{j>=0} 10^j*z(j)^(9/8)*(1 - 10z(j)^9 + 9z(j)^10)/((1-z(j))(1-z(j)^9)), where z(j) = x^9^j.
Also: g(x) = (1/(1-x)) Sum_{j>=0} (1 - 10(x^9^j)^9 + 9(x^9^j)^10)*x^9^j*f_j(x)/(1-x^9^j), where f_j(x) = 10^j*x^((9^j-1)/8)/(1-(x^9^j)^9). Here, the f_j obey the recurrence f_0(x) = 1/(1-x^9), f_(j+1)(x) = 10x*f_j(x^9).
Also: g(x) = (1/(1-x))*((Sum{k=0..8} h_(9,k)(x)) - 9*h_(9,9)(x)), where h_(9,k)(x) = Sum_{j>=0} 10^j*x^((9^(j+1)-1)/8)*x^(k*9^j)/(1-x^9^(j+1)).
Generic formulas for analogous sequences with numbers expressed in base p and only using the digits 1, 2, 3, ... d, where 1 < d < p:
a(n) = Sum_{j=0..m-1} (1 + b(j) mod d)*p^j, where m = floor(log_d((d-1)*n+1)), b(j) = floor(((d-1)*n+1-d^m)/((d-1)*d^j)).
Special values:
a(k*(d^n-1)/(d-1)) = k*(10^n-1)/9, k=1..d.
a(d*((2d-1)*d^(n-1)-1)/(d-1)) = ((d+9)*10^n-d)/9 = 10^n + d*(10^n-1)/9.
a((d^n-1)/(d-1)-1) = d*(10^(n-1)-1)/9, n > 1.
Inequalities:
a(n) <= (10^log_d((d-1)*n+1)-1)/9, equality holds for n = (d^k-1)/(d-1), k > 0.
a(n) > (d/10)*(10^log_d((d-1)*n+1)-1)/9, n > 0.
Lower and upper limits:
lim inf a(n)/10^log_d((d-1)*n) = d/90, for n -> infinity.
lim sup a(n)/10^log_d((d-1)*n) = 1/9, for n -> infinity.
G.f.: g(x) = (1/(1-x)) Sum_{j>=0} (1 - (d+1)(x^d^j)^d + d(x^d^j)^(d+1))*x^d^j*f_j(x)/(1-x^d^j), where f_j(x) = p^j*x^((d^j-1)/(d-1))/(1-(x^d^j)^d). Here, the f_j obey the recursion f_0(x) = 1/(1-x^d), f_(j+1)(x) = px*f_j(x^d).
(End)
A052382 = { n | A054054(n) > 0 }. - M. F. Hasler, Jan 23 2013
From Hieronymus Fischer, Feb 20 2019: (Start)
Sum_{n>=1} (-1)^(n+1)/a(n) = 0.696899720...
Sum_{n>=1} 1/a(n)^2 = 1.6269683705819...
Sum_{n>=1} 1/a(n) = 23.1034479... = A082839. This so-called Kempner series converges very slowly. For the calculation of the sum, it is helpful to use the following fraction of partial sums, which converges rapidly:
lim_{n->infinity} (Sum_{k=p(n)..p(n+1)-1} 1/a(k)) / (Sum_{k=p(n-1)..p(n)-1} 1/a(k)) = 9/10, where p(n) = (9^n-1)/8, n > 1.
(End)

Typos in formula section corrected by Hieronymus Fischer, May 30 2012

Name clarified by Peter Luschny, Jun 20 2025

A011531 Numbers that contain a digit 1 in their decimal representation.

Original entry on oeis.org

1, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 31, 41, 51, 61, 71, 81, 91, 100, 101, 102, 103, 104, 105, 106, 107, 108, 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

Offset: 1

N. J. A. Sloane

A121042(a(n)) = 1. - Reinhard Zumkeller, Jul 21 2006

See A043493 for numbers that contain a single digit '1'. A subsequence of numbers having a digit that divides all other digits, A263314. - M. F. Hasler, Jan 11 2016

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.

Cf. A121041, A121022, A121023, A121024, A121025, A121026, A121027, A121028, A121029, A121030, A121031, A121032, A121033, A121034, A121035, A121036, A121037, A121038, A121039, A121040.

Cf. A052383 (complement), A062634 (a subsequence).

Filtered([1..140],n->1 in ListOfDigits(n)); # Muniru A Asiru, Feb 23 2019

a011531 n = a011531_list !! (n-1)
a011531_list = filter ((elem '1') . show) [0..]
-- Reinhard Zumkeller, Feb 05 2012

[n: n in [0..500] | 1 in Intseq(n) ]; // Vincenzo Librandi, Jan 11 2016

M:= 3: # to get all terms of up to M digits
B:= {1}: A:= {1}:
for i from 2 to M do
   B:= map(t -> seq(10*t+j,j=0..9),B) union
      {seq(10*x+1,x=2*10^(i-2)..10^(i-1)-1)}:
   A:= A union B;
od:
sort(convert(A,list)); # Robert Israel, Jan 10 2016
# second program:
A011531 := proc(n)
    if n = 1 then
        1;
    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

Select[Range[600] - 1, DigitCount[#, 10, 1] > 0 &] (* Vincenzo Librandi, Jan 11 2016 *)

is_A011531(n)=setsearch(Set(digits(n)),1) \\ M. F. Hasler, Jan 10 2016

def aupto(nn): return [m for m in range(1, nn+1) if '1' in str(m)]
print(aupto(133)) # Michael S. Branicky, Jan 10 2021

(0 to 119).filter(.toString.indexOf('1') > -1) // _Alonso del Arte, Jan 12 2020

a(n) ~ n. - Charles R Greathouse IV, Nov 02 2022

A038603 Primes not containing the digit '1'.

Original entry on oeis.org

2, 3, 5, 7, 23, 29, 37, 43, 47, 53, 59, 67, 73, 79, 83, 89, 97, 223, 227, 229, 233, 239, 257, 263, 269, 277, 283, 293, 307, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 409, 433, 439, 443, 449, 457, 463, 467, 479, 487, 499, 503, 509, 523, 547, 557

Offset: 1

Vasiliy Danilov (danilovv(AT)usa.net), Jul 15 1998

Subsequence of A132080. - Reinhard Zumkeller, Aug 09 2007

Maynard proves that this sequence is infinite and in particular contains the expected number of elements up to x, on the order of x^(log 9/log 10)/log x. - Charles R Greathouse IV, Apr 08 2016

Indranil Ghosh, Table of n, a(n) for n = 1..50000 (terms 1..1000 from R. Zumkeller)
M. F. Hasler, Numbers avoiding certain digits, OEIS Wiki, Jan 12 2020.
James Maynard, Primes with restricted digits, arXiv:1604.01041 [math.NT], 2016.
James Maynard and Brady Haran, Primes without a 7, Numberphile video (2019).
Index to entries for primes with digits in a given set

Intersection of A000040 (primes) and A052383 (numbers with no digit 1).
Primes having no digit d = 0..9 are A038618, this sequence, A038604, A038611, A038612, A038613, A038614, A038615, A038616, and A038617, respectively.
Primes with other restrictions on digits: A106116, A156756.

[ p: p in PrimesUpTo(600) | not 1 in Intseq(p) ];  // Bruno Berselli, Aug 08 2011

Select[Prime[Range[70]], DigitCount[#, 10, 1] == 0 &] (* Vincenzo Librandi, Aug 09 2011 *)

is(n)=if(isprime(n),n=vecsort(eval(Vec(Str(n))),,8);n[1]>1||(!n[1]&&n[2]>1)) \\ Charles R Greathouse IV, Aug 09 2011

is(n)=!vecsearch(vecsort(digits(n)),1) && isprime(n) \\ Charles R Greathouse IV, Oct 03 2012

next_A038603(n)=until((n=nextprime(n+1))==n=next_A052383(n-1),);n \\ Compute least a(k) > n. See A052383. - M. F. Hasler, Jan 14 2020

from sympy import nextprime
i=p=1
while i<=500:
    p = nextprime(p)
    if '1' not in str(p):
        print(str(i)+" "+str(p))
        i+=1
# Indranil Ghosh, Feb 07 2017, edited by M. F. Hasler, Jan 15 2020
# See the OEIS Wiki page for more efficient programs. - M. F. Hasler, Jan 14 2020

a(n) ≍ n^(log 10/log 9) log n. - Charles R Greathouse IV, Aug 03 2023

A052405 Numbers without 3 as a digit.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 79, 80, 81, 82, 84, 85, 86, 87, 88, 89

Offset: 1

Henry Bottomley, Mar 13 2000

This sequence also represents the minimal number of straight lines of a covering tree to cover n X n points arranged in a symmetrical grid. - Marco Ripà, Sep 20 2018

			22 has no 3s among its digits, hence it is in the sequence.
23 has one 3 among its digits, hence it is not in the sequence.

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. A004178, A004722, A038611 (subset of primes), A082832 (Kempner series).
Cf. A052382 (without 0), A052383 (without 1), A052404 (without 2), A052406 (without 4), A052413 (without 5), A052414 (without 6), A052419 (without 7), A052421 (without 8), A007095 (without 9).
Cf. A011533 (complement).

a052405 = f . subtract 1 where
   f 0 = 0
   f v = 10 * f w + if r > 2 then r + 1 else r  where (w, r) = divMod v 9
-- Reinhard Zumkeller, Oct 07 2014

[ n: n in [0..89] | not 3 in Intseq(n) ];  // Bruno Berselli, May 28 2011

a:= proc(n) local l, m; l, m:= 0, n-1;
      while m>0 do l:= (d->
        `if`(d<3, d, 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

Select[Range[0, 89], DigitCount[#, 10, 3] == 0 &] (* Alonso del Arte, Oct 16 2012 *)

is(n)=n=digits(n);for(i=1,#n,if(n[i]==3,return(0)));1 \\ Charles R Greathouse IV, Oct 16 2012
apply( {A052405(n)=fromdigits(apply(d->d+(d>2),digits(n-1,9)))}, [1..99]) \\ a(n)
next_A052405(n, d=digits(n+=1))={for(i=1, #d, d[i]==3&& return((1+n\d=10^(#d-i))*d)); n} \\ least a(k) > n. Used in A038611. \\ M. F. Hasler, Jan 11 2020

from gmpy2 import digits
def A052405(n): return int(digits(n-1,9).translate(str.maketrans('345678','456789'))) # Chai Wah Wu, Jun 28 2025

seq 0 1000 | grep -v 3; # Joerg Arndt, May 29 2011

a(n) >> n^k with k = log(10)/log(9) = 1.0479.... - Charles R Greathouse IV, Oct 16 2012
a(n) = replace digits d > 2 by d + 1 in base-9 representation of n - 1. - Reinhard Zumkeller, Oct 07 2014
Sum_{n>1} 1/a(n) = A082832 = 20.569877... (Kempner series). - Bernard Schott, Jan 12 2020, edited by M. F. Hasler, Jan 14 2020

Offset changed by Reinhard Zumkeller, Oct 07 2014

A052413 Numbers without 5 as a digit.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83, 84, 86, 87, 88, 89

Offset: 1

Henry Bottomley, Mar 13 2000

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. A004180, A004724, A038613 (subset of primes), A082834 (Kempner series).

Cf. A052382 (without 0), A052383 (without 1), A052404 (without 2), A052405 (without 3), A052406 (without 4), A052414 (without 6), A052419 (without 7), A052421 (without 8), A007095 (without 9).

a052413 = f . subtract 1 where
f 0 = 0
f v = 10 * f w + if r > 4 then r + 1 else r where (w, r) = divMod v 9
-- Reinhard Zumkeller, Oct 07 2014

[ n: n in [0..89] | not 5 in Intseq(n) ]; // Bruno Berselli, May 28 2011

a:= proc(n) local l, m; l, m:= 0, n-1;
      while m>0 do l:= (d->
        `if`(d<5, d, 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

Select[Range[100],!MemberQ[IntegerDigits[#],5]&] (* Harvey P. Dale, Feb 20 2013 *)

apply( {A052413(n)=fromdigits(apply(d->d+(d>4),digits(n-1,9)))}, [1..99]) \\ a(n)
select( {is_A052413(n)=!setsearch(Set(digits(n)),5)}, [0..99]) \\ used in A038613
next_A052413(n, d=digits(n+=1))={for(i=1,#d, d[i]==5&&return((1+n\d=10^(#d-i))*d)); n} \\ least a(k) > n; used in A038613. - M. F. Hasler, Jan 11 2020

# see the OEIS wiki page (cf. LINKS) for more programs
def A052413(n): n-=1; return sum(n//9**e%9*6//5*10**e for e in range(math.ceil(math.log(n+1,9)))) # M. F. Hasler, Jan 13 2020

from gmpy2 import digits
def A052413(n): return int(digits(n-1,9).translate(str.maketrans('5678','6789'))) # Chai Wah Wu, Jun 28 2025

seq 0 1000 | grep -v 5; # Joerg Arndt, May 29 2011

a(n) = replace digits d > 4 by d + 1 in base-9 representation of n - 1. - Reinhard Zumkeller, Oct 07 2014

Sum_{k>1} 1/a(n) = A082834 = 21.8346008... (Kempner series). - Bernard Schott, Jan 12 2020, edited by M. F. Hasler, Jan 13 2020

Offset changed by Reinhard Zumkeller, Oct 07 2014

A052414 Numbers without 6 as a digit.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 82, 83, 84, 85, 87, 88, 89

Offset: 1

Henry Bottomley, Mar 13 2000

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. A004181, A004725, A038614 (subset of primes), A082835 (Kempner series).

Cf. A052382 (without 0), A052383 (without 1), A052404 (without 2), A052405 (without 3), A052406 (without 4), A052413 (without 5), A052419 (without 7), A052421 (without 8), A007095 (without 9).

a052414 = f . subtract 1 where
f 0 = 0
f v = 10 * f w + if r > 5 then r + 1 else r where (w, r) = divMod v 9
-- Reinhard Zumkeller, Oct 07 2014

[ n: n in [0..89] | not 6 in Intseq(n) ]; // Bruno Berselli, May 28 2011

a:= proc(n) local l, m; l, m:= 0, n-1;
      while m>0 do l:= (d->
        `if`(d<6, d, 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

Select[Range[0,100],DigitCount[#,10,6]==0&] (* Harvey P. Dale, Jun 20 2013 *)

lista(nn)=for (n=0, nn, if (!vecsearch(vecsort(digits(n),,8), 6), print1(n, ", "));); \\ Michel Marcus, Feb 22 2015

/* See OEIS wiki page (cf. LINKS) for more programs */
apply( {A052414(n)=fromdigits(apply(d->d+(d>5),digits(n-1,9)))}, [1..99]) \\ a(n)
select( {is_A052414(n)=!setsearch(Set(digits(n)),6)}, [0..99]) \\ used in A038614
next_A052414(n, d=digits(n+=1))={for(i=1,#d, d[i]==6&&return((1+n\d=10^(#d-i))*d)); n} \\ least a(k) > n, used in A038614. - M. F. Hasler, Jan 11 2020

from gmpy2 import digits
def A052414(n): return int(digits(n-1,9).translate(str.maketrans('678','789'))) # Chai Wah Wu, Jun 28 2025

seq 0 1000 | grep -v 6; # Joerg Arndt, May 29 2011

a(n) = replace digits d > 5 by d + 1 in base-9 representation of n - 1. - Reinhard Zumkeller, Oct 07 2014

Sum_{k>1} 1/a(k) = A082835 = 22.205598... (Kempner series). - Bernard Schott, Jan 12 2020, edited by M. F. Hasler, Jan 13 2020

Offset changed by Reinhard Zumkeller, Oct 07 2014

A052419 Numbers without 7 as a digit.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 80, 81, 82, 83, 84, 85, 86, 88, 89

Offset: 1

Henry Bottomley, Mar 13 2000

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. A004182, A004726, A038615 (subset of primes), A082836 (Kempner series).

Cf. A052382 (without 0), A052383 (without 1), A052404 (without 2), A052405 (without 3), A052406 (without 4), A052413 (without 5), A052414 (without 6), A052421 (without 8), A007095 (without 9).

a052419 = f . subtract 1 where
f 0 = 0
f v = 10 * f w + if r > 6 then r + 1 else r where (w, r) = divMod v 9
-- Reinhard Zumkeller, Oct 07 2014

[ n: n in [0..89] | not 7 in Intseq(n) ]; // Bruno Berselli, May 28 2011

a:= proc(n) local l, m; l, m:= 0, n-1;
      while m>0 do l:= (d->
        `if`(d<7, d, 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

Select[Range[100],DigitCount[#,10,7]==0&] (* Harvey P. Dale, Aug 23 2011 *)

lista(nn)=for (n=0, nn, if (!vecsearch(vecsort(digits(n),,8), 7), print1(n, ", "));); \\ Michel Marcus, Feb 22 2015

/* See OEIS wiki page for more programs. */
apply( {A052419(n)=fromdigits(apply(d->d+(d>6),digits(n-1,9)))}, [1..99]) \\ a(n)
select( {is_A052419(n)=!setsearch(Set(digits(n)),7)}, [0..99]) \\ used in A038615
next_A052419(n, d=digits(n+=1))={for(i=1,#d, d[i]==7&&return((1+n\d=10^(#d-i))*d)); n} \\ least a(k) > n. Used in A038615. - M. F. Hasler, Jan 11 2020

from gmpy2 import digits
def A052419(n): return int(digits(n-1,9).replace('8','9').replace('7','8')) # Chai Wah Wu, Jun 28 2025

seq 0 1000 | grep -v 7; # Joerg Arndt, May 29 2011

a(n) = replace digits d > 6 by d + 1 in base-9 representation of n - 1. - Reinhard Zumkeller, Oct 07 2014

Sum_{n>1} 1/a(n) = A082836 = 22.493475... (Kempner series). - Bernard Schott, Jan 12 2020, edited by M. F. Hasler, Jan 13 2020

Offset changed by Reinhard Zumkeller, Oct 07 2014

cp's OEIS Frontend

A248518 Number of partitions of n into parts > 0 without 1 as digit, cf. A052383.

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A248519 Number of partitions of n into distinct parts > 0 without 1 as digit, cf. A052383.

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A007095 Numbers in base 9.

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A052382 Numbers without 0 in the decimal expansion, colloquial 'zeroless numbers'.

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A011531 Numbers that contain a digit 1 in their decimal representation.

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A038603 Primes not containing the digit '1'.

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