A052419 -id:A052419 - 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

A082836 Decimal expansion of Kempner series Sum_{k >= 1, k has no digit 7 in base 10} 1/k.

Original entry on oeis.org

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

Offset: 2

Robert G. Wilson v, Apr 14 2003

Numbers with a digit 7 (A011537) have asymptotic density 1, i.e., here 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

			22.493475311705945398176226915339775974005915541672512361791460444... - _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. [_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. [_Robert G. Wilson v_, Jun 01 2009]

Cf. A002387, A024101, A052419 (numbers with no '7'), A011537 (numbers with a '7').

Cf. A082830, A082831, A082832, A082833, A082834, A082835, A082837, A082838, A082839 (analog for digits 1, 2, ..., 9 and 0).

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

Equals Sum_{k in A052419\{0}} 1/k, where A052419 = numbers with no digit 7. - M. F. Hasler, Jan 14 2020

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

Minor edits by M. F. Hasler, Jan 13 2020

A308262 Numbers m such that A048385(m) ends with m.

Original entry on oeis.org

0, 1, 5, 6, 10, 11, 25, 36, 50, 51, 60, 61, 100, 101, 110, 111, 250, 251, 360, 361, 425, 500, 501, 510, 511, 600, 601, 610, 611, 936, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 1936, 2500, 2501, 2510, 2511, 3600, 3601, 3610, 3611, 4250, 4251, 5000, 5001

Offset: 1

Rémy Sigrist, May 17 2019

If m belongs to this sequence, then A048385(m) belongs to this sequence.
If m belongs to this sequence, then 10*m and 10*m + 1 belong to this sequence.
This sequence contains A007088.
All terms belong to A052419.
Let U be the infinite word ...|A048385^2(16)|A048385(16)|16425 and V be the infinite word ...|A048385^2(81)|A048385(81)|81936. The terms of this sequence consist of the last x digits of either U or V followed by y digits in {0,1}, where x and y are nonnegative integers. - Charlie Neder, May 17 2019

			The first terms, alongside A048385(a(n)), are:
  n   a(n)  A048385(a(n))
  --  ----  -------------
   1     0              0
   2     1              1
   3     5             25
   4     6             36
   5    10             10
   6    11             11
   7    25            425
   8    36            936
   9    50            250
  10    51            251
  11    60            360
  12    61            361

Rémy Sigrist, PARI program for A308262

Cf. A003226, A007088, A048385, A052419.

m=1;
for u=0:5001
    digit=dec2base(u,10)-'0';digitp=digit.^2;
    aa=str2num(strrep(num2str(digitp), ' ', ''));
    digitaa=dec2base(aa,10)-'0';
       if mod(aa,10^length(digit))==u
        sol(m)=u; m=m+1;
       end
end
sol % Marius A. Burtea, May 17 2019

PARI
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See Links section.
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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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A082836 Decimal expansion of Kempner series Sum_{k >= 1, k has no digit 7 in base 10} 1/k.

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A308262 Numbers m such that A048385(m) ends with m.

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

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PARI

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

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