A032810 -id:A032810 - cp's OEIS frontend

A007931 Numbers that contain only 1's and 2's. Nonempty binary strings of length n in lexicographic order.

1, 2, 11, 12, 21, 22, 111, 112, 121, 122, 211, 212, 221, 222, 1111, 1112, 1121, 1122, 1211, 1212, 1221, 1222, 2111, 2112, 2121, 2122, 2211, 2212, 2221, 2222, 11111, 11112, 11121, 11122, 11211, 11212, 11221, 11222, 12111, 12112, 12121, 12122

Offset: 1

R. Muller

Numbers written in the dyadic system [Smullyan, Stillwell]. - N. J. A. Sloane, Feb 13 2019
Logic-binary sequence: prefix it by the empty word to have all binary words on the alphabet {1,2}.
The least binary word of length k is a(2^k - 1).
See Mathematica program for logic-binary sequence using (0,1) in place of (1,2); the sequence starts with 0,1,00,01,10. - Clark Kimberling, Feb 09 2012
A007953(a(n)) = A014701(n+1); A007954(a(n)) = A048896(n). - Reinhard Zumkeller, Oct 26 2012
a(n) is n written in base 2 where zeros are not allowed but twos are. The two distinct digits used are 1, 2 instead of 0, 1. To obtain this sequence from the "canonical" base 2 sequence with zeros allowed, just replace any 0 with a 2 and then subtract one from the group of digits situated on the left: (10-->2; 100-->12; 110-->22; 1000-->112; 1010-->122). - Robin Garcia, Jan 31 2014
For numbers made of only two different digits, see also A007088 (digits 0 & 1), A032810 (digits 2 & 3), A032834 (digits 3 & 4), A256290 (digits 4 & 5), A256291 (digits 5 & 6), A256292 (digits 6 & 7), A256340(digits 7 & 8), A256341 (digits 8 & 9), and A032804-A032816 (in other bases). Numbers with exactly two distinct (but unspecified) digits in base 10 are listed in A031955, for other bases in A031948-A031954. - M. F. Hasler, Apr 04 2015
The variant with digits {0, 1} instead of {1, 2} is obtained by deleting all initial digits in sequence A007088 (numbers written in base 2). - M. F. Hasler, Nov 03 2020

			Positive numbers may not start with 0 in the OEIS, otherwise this sequence would have been written as: 0, 1, 00, 01, 10, 11, 000, 001, 010, 011, 100, 101, 110, 111, 0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 00000, 00001, 00010, 00011, 00100, 00101, 00110, 00111, 01000, 01001, 01010, 01011, ...
From _Hieronymus Fischer_, Jun 06 2012: (Start)
a(10)   = 122.
a(100)  = 211212.
a(10^3) = 222212112.
a(10^4) = 1122211121112.
a(10^5) = 2111122121211112.
a(10^6) = 2221211112112111112.
a(10^7) = 11221112112122121111112.
a(10^8) = 12222212122221111211111112.
a(10^9) = 22122211221212211212111111112. (End)

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 2. - From N. J. A. Sloane, Jul 26 2012
K. Atanassov, On the 97th, 98th and the 99th Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 3, 89-93.
R. M. Smullyan, Theory of Formal Systems, Princeton, 1961.
John Stillwell, Reverse Mathematics, Princeton, 2018. See p. 90.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000 (terms up to 2^10-2 from T. D. Noe, corrected by Sean A. Irvine, April 18 2019)
K. Atanassov, On Some of Smarandache's Problems, American Research Press, 1999, 16-21.
EMIS, Mirror site for Southwest Journal of Pure and Applied Mathematics
R. R. Forslund, A logical alternative to the existing positional number system, Southwest Journal of Pure and Applied Mathematics, Vol. 1, 1995.
R. R. Forslund, Positive Integer Pages
James E. Foster, A Number System without a Zero-Symbol, Mathematics Magazine, Vol. 21, No. 1. (1947), pp. 39-41.
F. Smarandache, Only Problems, Not Solutions!.
Index entries for 10-automatic sequences.

Cf. A007932 (digits 1-3), A059893, A045670, A052382 (digits 1-9), A059939, A059941, A059943, A032924, A084544, A084545, A046034 (prime digits 2,3,5,7), A089581, A084984 (no prime digits); A001742, A001743, A001744: loops; A202267 (digits 0, 1 and primes), A202268 (digits 1,4,6,8,9), A014261 (odd digits), A014263 (even digits).
Cf. A007088 (digits 0 & 1), A032810 (digits 2 & 3), A032834 (digits 3 & 4), A256290 (digits 4 & 5), A256291 (digits 5 & 6), A256292 (digits 6 & 7), A256340 (digits 7 & 8), A256341 (digits 8 & 9), and A032804-A032816 (in other bases).
Cf. A020450 (primes).

a007931 n = f (n + 1) where
   f x = if x < 2 then 0 else (10 * f x') + m + 1
     where (x', m) = divMod x 2
-- Reinhard Zumkeller, Oct 26 2012

[n: n in [1..100000] | Set(Intseq(n)) subset {1,2}]; // Vincenzo Librandi, Aug 19 2016

# Maple program to produce the sequence:
a:= proc(n) local m, r, d; m, r:= n, 0;
      while m>0 do d:= irem(m, 2, 'm');
        if d=0 then d:=2; m:= m-1 fi;
        r:= d, r
      od; parse(cat(r))/10
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 26 2016
# Maple program to invert this sequence: given a(n), it returns n. - N. J. A. Sloane, Jul 09 2012
invert7931:=proc(u)
local t1,t2,i;
t1:=convert(u,base,10);
[seq(t1[i]-1,i=1..nops(t1))];
[op(%),1];
t2:=convert(%,base,2,10);
add(t2[i]*10^(i-1),i=1..nops(t2))-1;
end;

f[n_] := FromDigits[Rest@IntegerDigits[n + 1, 2] + 1]; Array[f, 42] (* Robert G. Wilson v Sep 14 2006 *)
(* Next, A007931 using (0,1) instead of (1,2) *)
d[n_] := FromDigits[Rest@IntegerDigits[n + 1, 2] + 1]; Array[FromCharacterCode[ToCharacterCode[ToString[d[#]]] - 1] &, 100] (* Peter J. C. Moses, at request of Clark Kimberling, Feb 09 2012 *)
Flatten[Table[FromDigits/@Tuples[{1,2},n],{n,5}]] (* Harvey P. Dale, Sep 13 2014 *)

apply( {A007931(n)=fromdigits([d+1|d<-binary(n+1)[^1]])}, [1..44]) \\ M. F. Hasler, Nov 03 2020, replacing older code from Mar 26 2015

/* inverse function */ apply( {A007931_inv(N)=fromdigits([d-1|d<-digits(N)],2)+2<M. F. Hasler, Nov 09 2020

def a(n): return int(bin(n+1)[3:].replace('1', '2').replace('0', '1'))
print([a(n) for n in range(1, 45)]) # Michael S. Branicky, May 13 2021

def A007931(n): return int(s:=bin(n+1)[3:])+(10**(len(s))-1)//9 # Chai Wah Wu, Jun 13 2025

To get a(n), write n+1 in base 2, remove initial 1, add 1 to all remaining digits: e.g., eleven (11) in base 2 is 1011; remove initial 1 and add 1 to remaining digits: a(10)=122. - Clark Kimberling, Mar 11 2003
Conversely, given a(n), to get n: subtract 1 from all digits, prefix with an initial 1, convert this binary number to base 10, subtract 1. E.g., a(6)=22 -> 11 -> 111 -> 7 -> 6. - N. J. A. Sloane, Jul 09 2012
a(n) = A053645(n+1)+A002275(A000523(n)) = a(n-2^b(n))+10^b(n) where b(n) = A059939(n) = floor(log_2(n+1)-1). - Henry Bottomley, Feb 14 2001
From Hieronymus Fischer, Jun 06 2012 and Jun 08 2012: (Start)
The formulas are designed to calculate base-10 numbers only using the digits 1 and 2.
a(n) = Sum_{j=0..m-1} (1 + b(j) mod 2)*10^j, where m = floor(log_2(n+1)), b(j) = floor((n+1-2^m)/(2^j)).
Special values:
a(k*(2^n-1)) = k*(10^n-1)/9, k= 1,2.
a(3*2^n-2) = (11*10^n-2)/9 = 10^n+2*(10^n-1)/9.
a(2^n-2) = 2*(10^(n-1)-1)/9, n>1.
Inequalities:
a(n) <= (10^log_2(n+1)-1)/9, equality holds for n=2^k-1, k>0.
a(n) > (2/10)*(10^log_2(n+1)-1)/9.
Lower and upper limits:
lim inf a(n)/10^log_2(n) = 1/45, for n --> infinity.
lim sup a(n)/10^log_2(n) = 1/9, for n --> infinity.
G.f.: g(x) = (1/(x(1-x)))*sum_{j=0..infinity} 10^j* x^(2*2^j)*(1 + 2 x^2^j)/(1 + x^2^j).
Also: g(x) = (1/(1-x))*(h_(2,0)(x) + h_(2,1)(x) - 2*h_(2,2)(x)), where h_(2,k)(x) = sum_{j>=0} 10^j*x^(2^(j+1)-1)*x^(k*2^j)/(1-x^2^(j+1)).
Also: g(x) = (1/(1-x)) sum_{j>=0} (1 - 3(x^2^j)^2 + 2(x^2^j)^3)*x^2^j*f_j(x)/(1-x^2^j), where f_j(x) = 10^j*x^(2^j-1)/(1-(x^2^j)^2). The f_j obey the recurrence f_0(x) = 1/(1-x^2), f_(j+1)(x) = 10x*f_j(x^2). (End)

Some crossrefs added by Hieronymus Fischer, Jun 06 2012

Edited by M. F. Hasler, Mar 26 2015

A020458 Primes that contain digits 2 and 3 only.

Original entry on oeis.org

2, 3, 23, 223, 233, 2333, 3323, 23333, 32233, 32323, 33223, 222323, 232333, 233323, 323233, 323333, 333233, 333323, 2222333, 2223233, 2232323, 2233223, 2332333, 2333323, 3222223, 3223223, 3223333, 3233323, 3233333, 3332233, 3333233, 22222223, 22223323, 22232233

Offset: 1

David W. Wilson

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
James Maynard and Brady Haran, Primes without a 7, Numberphile video (2019)

Cf. A139067, A139069, A032810, A020463.

Flatten[Table[Select[FromDigits/@Tuples[{2,3},n],PrimeQ],{n,7}]] (* Harvey P. Dale, Jul 13 2012 *)

go(n)=my(v=List([2]),x,t); for(d=1,n, x=10^d\9*2; forstep(i=1,2^d-1,2, if(ispseudoprime(t=x+fromdigits(binary(i))), listput(v,t)))); Vec(v) \\ Charles R Greathouse IV, Sep 14 2015

from sympy import isprime
from itertools import count, islice, product
def agen(): # generator of terms
    yield from [2, 3]
    for d in count(2):
        for first in product("23", repeat=d-1):
            t = int("".join(first) + "3")
            if isprime(t): yield t
print(list(islice(agen(), 34))) # Michael S. Branicky, Jun 08 2022

Edited by N. J. A. Sloane, Jul 27 2008 at the suggestion of Dmitry Kamenetsky.

Edited by Charles R Greathouse IV, Mar 17 2010

A256290 Numbers which have only digits 4 and 5 in base 10.

Original entry on oeis.org

4, 5, 44, 45, 54, 55, 444, 445, 454, 455, 544, 545, 554, 555, 4444, 4445, 4454, 4455, 4544, 4545, 4554, 4555, 5444, 5445, 5454, 5455, 5544, 5545, 5554, 5555, 44444, 44445, 44454, 44455, 44544, 44545, 44554, 44555, 45444, 45445, 45454, 45455, 45544

Offset: 1

M. F. Hasler, Mar 27 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..8190
Index entries for 10-automatic sequences.

Cf. A007088 (digits 0 & 1), A007931 (digits 1 & 2), A032810 (digits 2 & 3), A032834 (digits 3 & 4), A256291 (digits 5 & 6), A256292 (digits 6 & 7), A256340 (digits 7 & 8), A256341 (digits 8 & 9).

[n: n in [1..60000] | Set(IntegerToSequence(n, 10)) subset {5, 4}];

[n: n in [1..100000] | Set(Intseq(n)) subset {4,5}]; // Vincenzo Librandi, Aug 19 2016

Flatten[Table[FromDigits[#,10]&/@Tuples[{4,5},n],{n,5}]]

A256290(n)=vector(#n=binary(n+1)[2..-1],i,10^(#n-i))*n~+10^#n\9*4

def A256290(n): return int(bin(n+1)[3:])+(10**((n+1).bit_length()-1)-1<<2)//9 # Chai Wah Wu, Jul 15 2023

a(n) = A007931(n) + A002277(A000523(n+1)) = A032834(n) + A256077(n) etc.

A256291 Numbers which have only digits 5 and 6 in base 10.

Original entry on oeis.org

5, 6, 55, 56, 65, 66, 555, 556, 565, 566, 655, 656, 665, 666, 5555, 5556, 5565, 5566, 5655, 5656, 5665, 5666, 6555, 6556, 6565, 6566, 6655, 6656, 6665, 6666, 55555, 55556, 55565, 55566, 55655, 55656, 55665, 55666, 56555, 56556

Offset: 1

M. F. Hasler, Mar 27 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..8190
Index entries for 10-automatic sequences.

Cf. A007088 (digits 0 & 1), A007931 (digits 1 & 2), A032810 (digits 2 & 3), A032834 (digits 3 & 4), A256290 (digits 4 & 5), A256292 (digits 6 & 7), A256340 (digits 7 & 8), A256341 (digits 8 & 9).

[n: n in [1..60000] | Set(IntegerToSequence(n, 10)) subset {5, 6}];

[n: n in [1..100000] | Set(Intseq(n)) subset {5,6}]; // :Vincenzo Librandi_, Aug 19 2016

Flatten[Table[FromDigits[#,10]&/@Tuples[{5,6},n],{n,5}]]

A256291(n)=vector(#n=binary(n+1)[2..-1],i,10^(#n-i))*n~+10^#n\9*5

a(n) = A007931(n) + A002278(A000523(n+1)) = A256290(n) + A256077(n) etc.

A031955 Numbers with exactly two distinct base-10 digits.

Original entry on oeis.org

10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 100, 101, 110, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 131, 133, 141, 144, 151, 155, 161, 166

Offset: 1

Clark Kimberling

The three-digit terms are given by A210666(1,...,244). For numbers with exactly two distinct (but unspecified) digits in other bases, see A031948-A031954. For numbers made of two *given* digits, see A007088 (digits 0 & 1), A007931 (digits 1 & 2), A032810 (digits 2 & 3), A032834 (digits 3 & 4), A256290 (digits 4 & 5), A256291 (digits 5 & 6), A256292 (digits 6 & 7), A256340 (digits 7 & 8), A256341 (digits 8 & 9), and A032804-A032816 (in other bases). - M. F. Hasler, Apr 04 2015
A235154 is a subsequence. - Altug Alkan, Dec 03 2015
A235717 is a subsequence. - Robert Israel, Dec 03 2015

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

Different from A029742.

Cf. A043638, A101594, A031948, A031949, A031950, A031951, A031952, A031953, A031954, A235154, A235717.

a031955 n = a031955_list !! (n-1)
a031955_list = filter ((== 2) . a043537) [0..]
-- Reinhard Zumkeller, Feb 05 2012

M:= 5: # to get all terms < 10^M
sort([seq(seq(seq(seq(add(10^(m-j)*`if`(member(j,S2),d2,d1),j=1..m)  ,
  S2 = combinat:-powerset({$2..m}) minus {{}}),
  d2 = {$0..9} minus {d1}), d1 = 1..9), m=2..M)]); # Robert Israel, Dec 03 2015

Select[Range@ 166, Length@ Union@ IntegerDigits@ # == 2 &] (* Michael De Vlieger, Dec 03 2015 *)

is_A031955(n)=#Set(digits(n))==2 \\ M. F. Hasler, Apr 04 2015

def ok(n): return len(set(str(n))) == 2
print(list(filter(ok, range(167)))) # Michael S. Branicky, Oct 12 2021

A043537(a(n)) = 2. - Reinhard Zumkeller, Dec 03 2009

Name edited by Charles R Greathouse IV, Feb 13 2017

A032834 Numbers with digits 3 and 4 only.

Original entry on oeis.org

3, 4, 33, 34, 43, 44, 333, 334, 343, 344, 433, 434, 443, 444, 3333, 3334, 3343, 3344, 3433, 3434, 3443, 3444, 4333, 4334, 4343, 4344, 4433, 4434, 4443, 4444, 33333, 33334, 33343, 33344, 33433, 33434, 33443, 33444, 34333, 34334

Offset: 1

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for 10-automatic sequences.

Cf. A032829-A032833 (in other bases), A102659 (Lyndon words in this sequence), A007088 (digits 0 & 1), A007931 (digits 1 & 2), A032810 (digits 2 & 3), A256290 (digits 4 & 5), A256291 (digits 5 & 6), A256292 (digits 6 & 7), A256340 (digits 7 & 8), A256341 (digits 8 & 9).

[n: n in [1..35000] | Set(IntegerToSequence(n, 10)) subset {3, 4}]; // Vincenzo Librandi, May 30 2012

S[1]:= [3,4]:
for d from 2 to 5 do S[d]:= map(t -> (10*t+3,10*t+4), S[d-1]) od:
seq(op(S[d]),d=1..5); # Robert Israel, Apr 03 2017

Flatten[Table[FromDigits[#,10]&/@Tuples[{3,4},n],{n,5}]] (* Vincenzo Librandi, May 30 2012 *)

A032834(n)=vector(#n=binary(n+1)[2..-1],i,10^(#n-i))*n~+10^#n\3 \\ M. F. Hasler, Mar 27 2015

a(n) = A007931(n) + A002276(A000523(n+1)) = A032810(n) + A256077(n) etc. - M. F. Hasler, Mar 27 2015
From Robert Israel, Apr 03 2017: (Start)
a(2*n+1) = 10*a(n)+3.
a(2*n+2) = 10*a(n)+4.
G.f. g(x) satisfies g(x) = 10*(x+x^2)*g(x^2) + x*(3+4*x)/(1-x^2). (End)

Crossrefs added by M. F. Hasler, Mar 27 2015

Name corrected by Robert Israel, Apr 03 2017

A256292 Numbers which have only digits 6 and 7 in base 10.

Original entry on oeis.org

6, 7, 66, 67, 76, 77, 666, 667, 676, 677, 766, 767, 776, 777, 6666, 6667, 6676, 6677, 6766, 6767, 6776, 6777, 7666, 7667, 7676, 7677, 7766, 7767, 7776, 7777, 66666, 66667, 66676, 66677, 66766, 66767, 66776, 66777, 67666, 67667

Offset: 1

M. F. Hasler, Mar 27 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..8190
Index entries for 10-automatic sequences.

Cf. A007088 (digits 0 & 1), A007931 (digits 1 & 2), A032810 (digits 2 & 3), A032834 (digits 3 & 4), A256290 (digits 4 & 5), A256291 (digits 5 & 6), A256340 (digits 7 & 8), A256341 (digits 8 & 9).

[n: n in [1..35000] | Set(IntegerToSequence(n, 10)) subset {7, 6}];

[n: n in [1..100000] | Set(Intseq(n)) subset {6,7}]; // Vincenzo Librandi, Aug 19 2016

Flatten[Table[FromDigits[#,10]&/@Tuples[{6,7},n],{n,5}]]

A256292(n)=vector(#n=binary(n+1)[2..-1],i,10^(#n-i))*n~+10^#n\9*6

a(n) = A007931(n) + A002279(A000523(n+1)) = A256291(n) + A256077(n) etc.

A256340 Numbers which have only digits 7 and 8 in base 10.

Original entry on oeis.org

7, 8, 77, 78, 87, 88, 777, 778, 787, 788, 877, 878, 887, 888, 7777, 7778, 7787, 7788, 7877, 7878, 7887, 7888, 8777, 8778, 8787, 8788, 8877, 8878, 8887, 8888, 77777, 77778, 77787, 77788, 77877, 77878, 77887, 77888, 78777, 78778, 78787, 78788, 78877, 78878

Offset: 1

M. F. Hasler, Mar 27 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..8190
Index entries for 10-automatic sequences.

Cf. A007088 (digits 0 & 1), A007931 (digits 1 & 2), A032810 (digits 2 & 3), A032834 (digits 3 & 4), A256290 (digits 4 & 5), A256291 (digits 5 & 6), A256292 (digits 6 & 7), A256341 (digits 8 & 9).

[n: n in [1..35000] | Set(IntegerToSequence(n, 10)) subset {7, 8}];

[n: n in [1..100000] | Set(Intseq(n)) subset {7,8}]; // Vincenzo Librandi, Aug 19 2016

Flatten[Table[FromDigits[#,10]&/@Tuples[{7,8},n],{n,5}]]

A256340(n)=vector(#n=binary(n+1)[2..-1],i,10^(#n-i))*n~+10^#n\9*7

def a(n): return int(bin(n+1)[3:].replace('0', '7').replace('1', '8'))
print([a(n) for n in range(1, 45)]) # Michael S. Branicky, Jul 08 2021

a(n) = A007931(n) + A002280(A000523(n+1)) = A256292(n) + A256077(n) etc.

A185969 Let S be the sequence of power towers built of 2 and 3 sorted by their height and for equal heights - in lexicographic order: 2, 3, 2^2, 2^3, 3^2, 3^3, 2^2^2, 2^2^3 etc. A(n) = the permutation of indexes which reorders S by magnitude.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 6, 11, 8, 9, 12, 13, 15, 23, 10, 14, 19, 27, 16, 24, 17, 25, 20, 28, 21, 29, 31, 47, 39, 55, 18, 26, 22, 30, 35, 51, 43, 59, 32, 48, 40, 56, 33, 49, 41, 57, 36, 52, 44, 60, 37, 53, 45, 61, 63, 95, 79, 111, 71, 103, 87, 119, 34, 50, 42, 58, 38

Offset: 1

Vladimir Reshetnikov, Feb 07 2011

			a(6) =  7; tower(7)  = 2^2^2 = 2^4 =  16.
a(7) =  6; tower(6)  = 3^3   =        27.
a(8) = 11; tower(11) = 3^2^2 = 3^4 =  81.
a(9) =  8; tower(8)  = 2^2^3 = 2^8 = 256.

Alois P. Heinz, Table of n, a(n) for n = 1..14335
Vladimir Reshetnikov, 2-3 sequence puzzle, SeqFan list, Mar 18 2015
Index entries for sequences that are permutations of the natural numbers

Cf. A032810, A081241, A248907, A256179, A256231, A375374 (colexicographic instead of lexicographic order).

a(2*n-1) = A081241(2*A081241(a(n-1))+1) and a(2*n) = A081241(A081241(a(2*n-1))+1) for n >= 7. - Pontus von Brömssen, Aug 10 2024

More terms from Alois P. Heinz, Apr 05 2011

A256341 Numbers which have only digits 8 and 9 in base 10.

Original entry on oeis.org

8, 9, 88, 89, 98, 99, 888, 889, 898, 899, 988, 989, 998, 999, 8888, 8889, 8898, 8899, 8988, 8989, 8998, 8999, 9888, 9889, 9898, 9899, 9988, 9989, 9998, 9999, 88888, 88889, 88898, 88899, 88988, 88989, 88998, 88999, 89888, 89889

Offset: 1

M. F. Hasler, Mar 27 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..8190
Index entries for 10-automatic sequences.

Cf. A007088 (digits 0 & 1), A007931 (digits 1 & 2), A032810 (digits 2 & 3), A032834 (digits 3 & 4), A256290 (digits 4 & 5) - A256292 (digits 6 & 7), A256340 (digits 7 & 8).

[n: n in [1..35000] | Set(IntegerToSequence(n, 10)) subset {8, 9}];

[n: n in [1..100000] | Set(Intseq(n)) subset {8,9}]; // Vincenzo Librandi, Aug 19 2016

Flatten[Table[FromDigits[#,10]&/@Tuples[{8,9},n],{n,5}]]

A256341(n)=vector(#n=binary(n+1)[2..-1],i,10^(#n-i))*n~+10^#n\9*8

def a(n): return int(bin(n+1)[3:].replace('0', '8').replace('1', '9'))
print([a(n) for n in range(1, 45)]) # Michael S. Branicky, Aug 09 2021

a(n) = A007931(n) + A002281(A000523(n+1)) = A256341(n) + A256077(n) etc.

cp's OEIS Frontend

A007931 Numbers that contain only 1's and 2's. Nonempty binary strings of length n in lexicographic order.

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A020458 Primes that contain digits 2 and 3 only.

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A256290 Numbers which have only digits 4 and 5 in base 10.

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A256291 Numbers which have only digits 5 and 6 in base 10.

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A031955 Numbers with exactly two distinct base-10 digits.

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A032834 Numbers with digits 3 and 4 only.

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