A052383 -id:A052383 - cp's OEIS frontend

A082830 Decimal expansion of Kempner series Sum_{k>=1, k has no digit 1 in base 10} 1/k.

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

Offset: 2

Robert G. Wilson v, Apr 14 2003

Such sums are called Kempner series, see A082839 (the analog for digit 0) for more information. - M. F. Hasler, Jan 13 2020

			16.17696952812344426657...

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. [From _R. J. Mathar_, Aug 07 2010]
Wikipedia, Kempner series.
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, A052383 (numbers without '1'), A011531 (numbers with '1').

Cf. A082831, A082832, A082833, A082834, A082835, A082836, A082837, A082838, A082839 (analog for digits 2, ..., 9 and 0).

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

Equals Sum_{k in A052383\{0}} 1/k, where A052383 = numbers with no digit 1. Those which have a digit 1 (A011531) are omitted in the harmonic sum, and they have asymptotic density 1: almost all terms are omitted from the sum. - M. F. Hasler, Jan 15 2020

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

A052406 Numbers without 4 as a digit.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89

Offset: 1

Henry Bottomley, Mar 13 2000

This is a frequent sequence on Chinese, Japanese and Korean elevator buttons. - Jean-Sebastien Girard (circeus(AT)hotmail.com), Jul 28 2008

Essentially numbers in base 9 (using digits 0, 1, 2, 3, 5, 6, 7, 8, 9 rather than 0-8). - Charles R Greathouse IV, Oct 13 2013

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
M. F. Hasler, Numbers avoiding certain digits OEIS wiki, Jan 12 2020.
Eric Weisstein's World of Mathematics, Uban Number
Wikipedia, Tetraphobia.
Index entries for 10-automatic sequences.

Cf. A004179, A004723, A011760, A038612 (subset of primes), A082833 (Kempner series).

Cf. A052382 (without 0), A052383 (without 1), A052404 (without 2), A052405 (without 3), A052413 (without 5), A052414 (without 6), A052419 (without 7), A052421 (without 8), A007095 (without 9).

a052406 = f . subtract 1 where
f 0 = 0
f v = 10 * f w + if r > 3 then r + 1 else r where (w, r) = divMod v 9
-- Reinhard Zumkeller, Oct 07 2014

[ n: n in [0..89] | not 4 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<4, 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, 4] == 0 &] (* Alonso del Arte, Oct 13 2013 *)

g(n)= local(x,v,j,flag); for(x=1,n, v=Vec(Str(x)); flag=1; for(j=1,length(v), if(v[j]=="4",flag=0)); if(flag,print1(x",") ) ) \\ Cino Hilliard, Apr 01 2007

apply( {A052406(n)=fromdigits(apply(d->d+(d>3),digits(n-1,9)))}, [1..99]) \\ a(n)
select( {is_A052406(n)=!setsearch(Set(digits(n)),4)}, [0..99]) \\ Used in A038612
next_A052406(n, d=digits(n+=1))={for(i=1, #d, d[i]!=4|| return((1+n\d=10^(#d-i))*d)); n} \\ least a(k) > n. Used for A038612. - M. F. Hasler, Jan 11 2020

def A052406(n): n-=1; return sum((d+(d>3))*10**i for d,i in ((n//9**i%9,i) for i in range(math.ceil(math.log(n+1,9))))) # M. F. Hasler, Jan 13 2020

from gmpy2 import digits
def A052406(n): return int(digits(n-1,9).translate(str.maketrans('45678','56789'))) # Chai Wah Wu, Jun 28 2025

seq 0 1000 | grep -v 4; # Joerg Arndt, May 29 2011

a(n) = replace digits d > 3 by d + 1 in base-9 representation of n - 1. - Reinhard Zumkeller, Oct 07 2014

Sum_{k>1} 1/a(k) = A082833 = 21.327465... (Kempner series). - Bernard Schott, Jan 12 2020, edited by M. F. Hasler, Jan 13 2020

Offset changed by Reinhard Zumkeller, Oct 07 2014

A052404 Numbers without 2 as a digit.

Original entry on oeis.org

0, 1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 83, 84, 85, 86, 87, 88, 89

Offset: 1

Henry Bottomley, Mar 13 2000

M. J. Halm, Word Weirdness, Mpossibilities 66 (Feb. 1998), p. 5.

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

Cf. A004177, A004721, A072809, A082831 (Kempner series).
Cf. A052382 (without 0), A052383 (without 1), A052405 (without 3), A052406 (without 4), A052413 (without 5), A052414 (without 6), A052419 (without 7), A052421 (without 8), A007095 (without 9).
See A038604 for the subset of primes. - M. F. Hasler, Jan 11 2020

a052404 = f . subtract 1 where
   f 0 = 0
   f v = 10 * f w + if r > 1 then r + 1 else r  where (w, r) = divMod v 9
-- Reinhard Zumkeller, Oct 07 2014

[ n: n in [0..89] | not 2 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<2, 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

ban2Q[n_]:=FreeQ[IntegerDigits[n],2]==True; Select[Range[0,89],ban2Q[#] &] (* Jayanta Basu, May 17 2013 *)
Select[Range[0,100],DigitCount[#,10,2]==0&] (* Harvey P. Dale, Apr 13 2015 *)

lista(nn, d=2) = {for (n=0, nn, if (!vecsearch(vecsort(digits(n),,8),d), print1(n, ", ")););} \\ Michel Marcus, Feb 21 2015

apply( {A052404(n)=fromdigits(apply(d->d+(d>1),digits(n-1,9)))}, [1..99])
next_A052404(n, d=digits(n+=1))={for(i=1, #d, d[i]==2&&return((1+n\d=10^(#d-i))*d)); n} \\ least a(k) > n: if there's a digit 2 in n+1, replace the first occurrence by 3 and all following digits by 0.
(A052404_vec(N)=vector(N, i, N=if(i>1, next_A052404(N))))(99) \\ first N terms
select( {is_A052404(n)=!setsearch(Set(digits(n)),2)}, [0..99])
(A052404_upto(N)=select( is_A052404, [0..N]))(99) \\ M. F. Hasler, Jan 11 2020

from gmpy2 import digits
def A052404(n): return int(''.join(str(int(d)+1) if d>'1' else d for d in digits(n-1,9))) # Chai Wah Wu, Aug 30 2024

seq 0 1000 | grep -v 2; # Joerg Arndt, May 29 2011

If the offset were changed to 0: a(0) = 0, a(n+1) = f(a(n)+1,a(n)+1) where f(x,y) = if x<10 and x<>2 then y else if x mod 10 = 2 then f(y+1,y+1) else f(floor(x/10),y). - Reinhard Zumkeller, Mar 02 2008
a(n) = replace digits d > 1 by d + 1 in base-9 representation of n - 1. - Reinhard Zumkeller, Oct 07 2014
Sum_{k>1} 1/a(k) = A082831 = 19.257356... (Kempner series). - Bernard Schott, Jan 12 2020, edited by M. F. Hasler, Jan 14 2020

Offset changed by Reinhard Zumkeller, Oct 07 2014

A052421 Numbers without 8 as a digit.

Original entry on oeis.org

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

A178333 Characteristic function of mountain numbers.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Reinhard Zumkeller, May 25 2010

a(A134941(n)) = 1; a(A134951(n)) = 1;
a(n) = 0 for n > 12345678987654321;
a(A011540(n))=0; a(A052383(n))=0; a(A171901(n))=0;
A178334(n) = SUM(a(k): 0<=k<=n).

R. Zumkeller, Table of n, a(n) for n = 0..20000
Index entries for characteristic functions

Cf. A000030, A196368.

a178333 n = fromEnum $
   n `mod` 10 == 1 && a000030 n == 1 && a196368 n == 1 && and down where
      down = dropWhile (== False) $ zipWith (<) (tail $ show n) (show n)
a178333_list = map a178333 [0..]
-- Reinhard Zumkeller, Oct 28 2001

a[n_] := Boole[ MatchQ[ IntegerDigits[n], {1, a___, b_, c___, 1} /; OrderedQ[{1, a, b}, Less] && OrderedQ[ {b, c, 1}, Greater]]]; a[1]=1; Table[a[n], {n, 0, 200}] (* Jean-François Alcover, Jun 13 2012 *)

a(n) = if n mod 10 = 1 then if n = 1 then 1 else g(n div 10, 1) else 0
with g(x, y) = if x mod 10 > y then g(x div 10, x mod 10) else if x mod 10 = y then 0 else h(x div 10, x mod 10)
and h(x, y) = if y = 1 then 0^x else if x mod 10 < y then h(x div 10, x mod 10) else 0.

A255398 Numbers k such that k^2 lacks the digit 1 in its decimal expansion.

Original entry on oeis.org

0, 2, 3, 5, 6, 7, 8, 15, 16, 17, 18, 20, 22, 23, 24, 25, 26, 27, 28, 30, 45, 47, 48, 50, 52, 53, 55, 57, 58, 60, 62, 63, 64, 65, 66, 67, 68, 70, 73, 74, 75, 76, 77, 78, 80, 82, 83, 84, 85, 86, 87, 88, 92, 93, 94, 95, 97, 98, 143, 144, 150, 153, 155, 156, 157, 158

Offset: 1

Vincenzo Librandi, Feb 22 2015

			98 is in this sequence because 98^2 = 9604.
99 is not in this sequence because 99^2 = 9801.

Mohammed Yaseen, Table of n, a(n) for n = 1..10000

Cf. A052383, A052040, A099150.

[n: n in [0..200] | not 1 in Intseq(n^2)];

filter:= n -> not member(1, convert(n^2,base,10)):
select(filter, [$0..200]); # Robert Israel, Apr 27 2023

Select[Range[0, 200], DigitCount[#^2, 10, 1]==0 &]

isok(k) = !vecsearch(Set(digits(k^2)), 1); \\ Michel Marcus, Apr 29 2023

def ok(k): return "1" not in str(k**2)
print([k for k in range(160) if ok(k)]) # Michael S. Branicky, Apr 27 2023

From Mohammed Yaseen, Apr 18 2023: (Start)
The smallest n-digit term ~ sqrt(2) * 10^(n-1).
The largest n-digit term = 10^n - 2 (see A099150). (End)

A299690 Numbers without digit 1 whose multiplicative digital root is not 0.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 22, 23, 24, 26, 27, 28, 29, 32, 33, 34, 35, 36, 37, 38, 39, 42, 43, 44, 46, 47, 48, 49, 53, 57, 62, 63, 64, 66, 67, 68, 72, 73, 74, 75, 76, 77, 79, 82, 83, 84, 86, 88, 89, 92, 93, 94, 97, 98, 99, 222, 223, 224, 226, 227, 228, 229, 232

Offset: 1

J. Lowell, Feb 19 2018

Is this sequence infinite?
There are no members of this sequence with 45 to 2000 decimal digits. Perhaps the last term is a(614640917006263790) = 77333222222222222222222222222222222222222222. - Charles R Greathouse IV, Feb 26 2018
This sequence is finite. Proof: Let k be the smallest term with more than 2000 decimal digits. Then the product of decimal digits pk of k has fewer than 2001 decimal digits (otherwise k isn't the smallest term with more than 2000 decimal digits). This number pk has at least as many decimal digits as 2^2001 has, which are 603. But then it doesn't have a nonzero multiplicative digital root per the computations of Charles R Greathouse IV. QED. - David A. Corneth, Aug 23 2018

			5 times 4 = 20 and 2 times 0 = 0, so 54 is not in this sequence.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A031347, A052383, A277061.

multDigRoot[n_] := NestWhile[Times @@ IntegerDigits@# &, n, UnsameQ, All]; Select[Range[500], DigitCount[#, 10, 1] == 0 && multDigRoot[#] != 0 &] (* Alonso del Arte, Feb 19 2018, based on Robert G. Wilson v's program for A031347 *)

mdr(n)=while(n>9,n=factorback(digits(n)));n
do(n)=my(v=List());forvec(u=vector(n,i,[2,9]), if(mdr(factorback(u)), listput(v, fromdigits(u)))); Vec(v) \\ Gives n-digit elements
\\ Charles R Greathouse IV, Feb 19 2018

{ A052383 } intersect { A277061 }.

A132080 Numbers having in decimal representation no common digits with all their proper divisors.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 23, 27, 29, 34, 37, 38, 43, 46, 47, 49, 53, 54, 56, 57, 58, 59, 67, 68, 69, 73, 76, 78, 79, 83, 86, 87, 89, 97, 223, 227, 229, 233, 239, 247, 249, 257, 259, 263, 267, 269, 277, 283, 289, 293, 307, 323, 329, 334, 337, 347, 349, 353, 356, 358, 359

Offset: 1

Reinhard Zumkeller, Aug 09 2007

A038603 is a subsequence.

			Proper divisors of a(20)=54: {1, 2, 3, 6, 9, 18, 27}
with set of digits {1,2,3,6,7,8,9} containing neither 4 nor 5.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Proper Divisor

Subsequence of A052383.

import Data.List (intersect)
a132080 n = a132080_list !! (n-1)
a132080_list = [x | x <- [2..], all null $
                    map (show x `intersect`) $ map show $ a027751_row x]
-- Reinhard Zumkeller, Oct 06 2012

Select[Range[2,359],ContainsNone[IntegerDigits[#],IntegerDigits/@Drop[Divisors[#],-1]//Flatten]&] (* James C. McMahon, Mar 03 2025 *)

is(n)=my(D=Set(digits(n))); fordiv(n,d, if(#setintersect(D, Set(digits(d))), return(d==n&&n>1))) \\ Charles R Greathouse IV, Dec 01 2013

A255430 Natural numbers n, not multiples of 10, such that n and n^2 lack the digit 1 in their decimal expansions.

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 22, 23, 24, 25, 26, 27, 28, 45, 47, 48, 52, 53, 55, 57, 58, 62, 63, 64, 65, 66, 67, 68, 73, 74, 75, 76, 77, 78, 82, 83, 84, 85, 86, 87, 88, 92, 93, 94, 95, 97, 98, 202, 205, 206, 207, 208, 222, 223, 225, 232, 233, 234, 235, 236, 238, 242, 243, 244, 245, 252, 253, 255

Offset: 1

Zak Seidov, Feb 23 2015

No trailing zeros allowed.

Cf. A000027, A052383, A067251, A255398.

A382239 Numbers not divisible by any of their digits nor by the sum of their digits. Digit 0 is allowed (and does not divide anything).

Original entry on oeis.org

23, 29, 34, 37, 38, 43, 46, 47, 49, 53, 56, 57, 58, 59, 67, 68, 69, 73, 74, 76, 78, 79, 83, 86, 87, 89, 94, 97, 98, 203, 223, 227, 229, 233, 239, 249, 253, 257, 259, 263, 267, 269, 277, 283, 289, 293, 299, 307, 323, 329, 334, 337, 338, 343, 346, 347, 349, 353, 356, 358, 359, 367, 373, 374, 376

Offset: 1

Robert Israel, Mar 19 2025

From a suggestion by Sergio Pimentel.

			a(5) = 38 is included because 38 is not divisible by 3, 8 or 3 + 8 = 11.
a(30) = 203 is included because 203 is not divisible by 2, 0, 3 or 2 + 0 + 3 = 5.

Robert Israel, Table of n, a(n) for n = 1..10000

Subsequence of A052383.

Cf. A038772, A082943, A382237.

filter:= proc(n) local L;
  L:= subs(0=NULL,convert(n,base,10));
  not ormap(t -> n mod t = 0, [op(L),convert(L,`+`)])
end proc:
select(filter, [$1..1000]);

s= {};Do[t=Select[IntegerDigits[n],#>0&];AppendTo[t,Total[t]];If[NoneTrue[t,Mod[n,#]==00&],AppendTo[s,n]],{n,376}];s (* James C. McMahon, Mar 21 2025 *)

def ok(n):
    d = list(map(int, str(n)))
    return (s:=sum(d)) and n%s!=0 and all(n%di!=0 for di in set(d)-{0})
print([k for k in range(1, 377) if ok(k)]) # Michael S. Branicky, Apr 01 2025

n^k << a(n) < 2^n for n > 5 where k = log(10)/log(9). - Charles R Greathouse IV, Mar 20 2025

cp's OEIS Frontend

A082830 Decimal expansion of Kempner series Sum_{k>=1, k has no digit 1 in base 10} 1/k.

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A052406 Numbers without 4 as a digit.

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A052404 Numbers without 2 as a digit.

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A052421 Numbers without 8 as a digit.

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A178333 Characteristic function of mountain numbers.

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A255398 Numbers k such that k^2 lacks the digit 1 in its decimal expansion.