A052405 -id:A052405 - cp's OEIS frontend

A052421 Numbers without 8 as a digit.

0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 79

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. A004183, A004727, A038616 (subset of primes), A082837 (Kempner series).

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), A007095 (without 9).

a052421 = f . subtract 1 where
f 0 = 0
f v = 10 * f w + if r > 7 then r + 1 else r where (w, r) = divMod v 9
-- Reinhard Zumkeller, Oct 07 2014

[ n: n in [0..89] | not 8 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<8, 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[#,10,8]==0&] (* Harvey P. Dale, Oct 11 2012 *)

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

/* See OEIS wiki page (cf. LINKS) for more programs. */
apply( {A052421(n)=fromdigits(apply(d->d+(d>7),digits(n-1,9)))}, [1..99]) \\ a(n)
select( {is_A052421(n)=!setsearch(Set(digits(n)),8)}, [0..99]) \\ used in A038616
next_A052421(n, d=digits(n+=1))={for(i=1,#d, d[i]==8&&return((1+n\d=10^(#d-i))*d)); n} \\ Least a(k) > n. Used in A038616. - M. F. Hasler, Jan 11 2020

from gmpy2 import digits
def A052421(n): return int(digits(n-1,9).replace('8','9')) # Chai Wah Wu, Jun 28 2025

seq 0 1000 | grep -v 8; # Joerg Arndt, May 29 2011

a(n) = replace digits d > 7 by d + 1 in base-9 representation of n - 1. - Reinhard Zumkeller, Oct 07 2014

Sum_{n>1} 1/a(n) = A082837 = 22.726365... (Kempner series). - Bernard Schott, Jan 12 2020, edited by M. F. Hasler, Jan 13 2020

Offset changed by Reinhard Zumkeller, Oct 07 2014

A082832 Decimal expansion of Sum_{k >= 1, k has no digit 3 in base 10} 1/k.

Original entry on oeis.org

2, 0, 5, 6, 9, 8, 7, 7, 9, 5, 0, 9, 6, 1, 2, 3, 0, 3, 7, 1, 0, 7, 5, 2, 1, 7, 4, 1, 9, 0, 5, 3, 1, 1, 1, 4, 1, 4, 1, 5, 3, 8, 6, 9, 6, 7, 4, 7, 3, 0, 7, 8, 3, 4, 8, 9, 5, 0, 8, 5, 2, 8, 5, 0, 0, 2, 6, 7, 2, 9, 4, 9, 9, 6, 1, 9, 3, 8, 0, 3, 5, 0, 0, 5, 9, 0, 4, 7, 4, 9, 4, 0, 8, 0, 6, 0, 3, 5, 3, 4, 9, 8, 7, 9, 0

Offset: 2

Robert G. Wilson v, Apr 14 2003

Numbers with a digit 3 (A011533) have asymptotic density 1, i.e., almost all terms are removed from the harmonic series, which makes convergence less surprising. See A082839 (the analog for digit 0) for more information about such so-called Kempner series. - M. F. Hasler, Jan 13 2020

			20.569877950961230371075217419053111414153869674730783489508528500... - _Robert G. Wilson v_, Jun 01 2009

Julian Havil, Gamma, Exploring Euler's Constant, Princeton University Press, Princeton and Oxford, 2003, page 34.

Robert Baillie, Sums of reciprocals of integers missing a given digit, Amer. Math. Monthly, 86 (1979), 372-374.
Robert Baillie, Summing the curious series of Kempner and Irwin, arXiv:0806.4410 [math.CA], 2008-2015. [From _Robert G. Wilson v_, Jun 01 2009]
Eric Weisstein's World of Mathematics, Kempner Series.
Wikipedia, Kempner series. [From _M. F. Hasler_, Jan 13 2020]
Wolfram Library Archive, KempnerSums.nb (8.6 KB) - Mathematica Notebook, Summing Kempner's Curious (Slowly-Convergent) Series. [From _Robert G. Wilson v_, Jun 01 2009]

Cf. A002387, A024101, A052405 (numbers with no '3'), A011533 (numbers with '3').

Cf. A082830, A082831, A082833, A082834, A082835, A082836, A082837, A082838, A082839 (analog for digits 1, 2, 4, ..., 9 and 0).

(* see the Mmca in Wolfram Library Archive. - Robert G. Wilson v, Jun 01 2009 *)

Equals Sum_{k in A052405\{0}} 1/k, where A052405 = numbers with no digit 3. - M. F. Hasler, Jan 15 2020

More terms from Robert G. Wilson v, Jun 01 2009

A235498 For k in {2,3,...,9} define a sequence as follows: a(0)=0; for n>=0, a(n+1)=a(n)+1, unless a(n) ends in k, in which case a(n+1) is obtained by replacing the last digit of a(n) with the digit(s) of k^2. This is k(2).

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, 30, 31, 32, 34, 35, 36, 37, 38, 39, 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

Offset: 0

Vincenzo Librandi, Feb 06 2014

An alternative and simpler definition: numbers not ending in 3. - Charles R Greathouse IV, Feb 07 2014

Subsequence of A052405.

Cf. A235499, A237341 - A237346.

G.f.: (x^9 +x^8 +x^7 +x^6 +x^5 +x^4 +2*x^3 +x^2 +x) / (x^10 -x^9 -x +1). - Alois P. Heinz, Feb 07 2014

Definition by N. J. A. Sloane, Feb 07 2014

A004178 Omit 3's from n.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

a(n) = 0 when n = 0 or when n is a 3-repdigit (A002277). a(n) = n when it contains no 3 among its digits. - Alonso del Arte, Oct 16 2012

			a(13) = 1 because after deleting the 3 from its base 10 representation, we're left with 1.
a(14) = 14 because there are no 3s to delete.

Cf. A004722, A052405.

Table[FromDigits[DeleteCases[IntegerDigits[n], 3]], {n, 0, 74}] (* Alonso del Arte, Oct 16 2012 *)

a(n)=subst(Pol(select(k->k-3,digits(n))),'x,10) \\ Charles R Greathouse IV, Oct 16 2012

A244008 Nonnegative integers with no repeated letters in their combined English decimal digit names.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 8, 16, 25, 26, 46, 48, 52, 60, 61, 62, 64, 84

Offset: 1

Rick L. Shepherd, Jun 17 2014

			The first multi-digit term is 16 since "one" and "six" taken together contain no duplicate letters. Although "one" itself contains no duplicate letters, by definition 11 is not a term since duplicate digits introduce repeated letters.

Subsequence of A010784, A052405, A052419, and A007095.

Cf. A059916.

is(n)=my(d=apply(k->[25,9,40,64,26,7,4,64,37,64][k+1], digits(n)),t); for(i=1,#d, if(bitand(t,d[i]), return(0)); t=bitor(t,d[i])); t<64 \\ Charles R Greathouse IV, Aug 18 2022

m = ["zero", "one", "two", "three", "four", "five", "six", "seven", "eight", "nine"]
def nr(w): return len(w) == len(set(w))
afull = [k for k in range(988) if nr("".join(m[int(d)] for d in str(k)))]
print(afull) # Michael S. Branicky, Aug 18 2022

A338287 Decimal expansion of the sum of reciprocals of the numbers that are not pandigital numbers (version 2, A171102).

Original entry on oeis.org

6, 5, 7, 4, 3, 3, 1, 1, 1, 0, 1, 8, 5, 3, 2, 8, 1, 9, 6, 7, 3, 4, 5, 8, 3, 1, 6, 7, 6, 8, 0, 8, 6, 8, 4, 1, 1, 6, 8, 5, 3, 4, 4, 1, 0, 6, 6, 3, 5, 3, 9, 8, 1, 6, 1, 0, 5, 0, 4, 3, 9, 2, 6, 3, 4, 6, 1, 3, 8, 7, 3, 8, 7, 3, 7, 1, 8, 5, 2, 6, 8, 0, 3, 4, 7, 8, 2

Offset: 2

Amiram Eldar, Oct 20 2020

The sum of the reciprocals of the terms of the complement of A171102: numbers with at most 9 distinct digits. It is the union of the 10 sequences of numbers without a single given digit (see the Crossrefs section).

The terms in the data section were taken from the 200 decimal digits given by Strich and Müller (2020).

			65.74331110185328196734583167680868411685344106635398...

Robert Strich and Eric Müller, Ziffernreduzierte Zahlen, in: E. Specht, E. Quaisser and P. Bauermann (eds.), 50 Jahre Bundeswettbewerb Mathematik, Springer Spektrum, Berlin, Heidelberg, 2020, pp. 171-182.

Cf. A171102, A179954, A082830, A082831, A082832, A082833, A082834, A082835, A082836, A082837, A082838, A082839.

Cf. A052382 (numbers without the digit 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), A007095 (without 9).

Equals 1/1 + 1/2 + 1/3 + ... + 1/1023456788 + 1/1023456790 + ..., i.e., A171102(1) = 1023456789 is the first number whose reciprocal is not in the sum.

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A052421 Numbers without 8 as a digit.

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A082832 Decimal expansion of Sum_{k >= 1, k has no digit 3 in base 10} 1/k.

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A235498 For k in {2,3,...,9} define a sequence as follows: a(0)=0; for n>=0, a(n+1)=a(n)+1, unless a(n) ends in k, in which case a(n+1) is obtained by replacing the last digit of a(n) with the digit(s) of k^2. This is k(2).

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A004178 Omit 3's from n.

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A244008 Nonnegative integers with no repeated letters in their combined English decimal digit names.

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Python

A338287 Decimal expansion of the sum of reciprocals of the numbers that are not pandigital numbers (version 2, A171102).

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