A007931 -id:A007931 - cp's OEIS frontend

A284293 Numbers using only digits 1 and 6.

1, 6, 11, 16, 61, 66, 111, 116, 161, 166, 611, 616, 661, 666, 1111, 1116, 1161, 1166, 1611, 1616, 1661, 1666, 6111, 6116, 6161, 6166, 6611, 6616, 6661, 6666, 11111, 11116, 11161, 11166, 11611, 11616, 11661, 11666, 16111, 16116, 16161, 16166, 16611, 16616

Offset: 1

Jaroslav Krizek, Mar 25 2017

Product of digits of n is a power of 6; subsequence of A276038.

Prime terms are in A020454.

Alois P. Heinz, Table of n, a(n) for n = 1..16382

Cf. Numbers using only digits 1 and k for k = 0 and k = 2 - 9: A007088 (k = 0), A007931 (k = 2), A032917 (k = 3), A032822 (k = 4) , A276037 (k = 5), this sequence (k = 6), A276039 (k = 7), A213084 (k = 8), A284294 (k = 9).

[n: n in [1..20000] | Set(IntegerToSequence(n, 10)) subset {1, 6}];

Join @@ (FromDigits /@ Tuples[{1,6}, #] & /@ Range[5]) (* Giovanni Resta, Mar 25 2017 *)

def A284293(n): return 5*int(bin(n+1)[3:])+(10**((n+1).bit_length()-1)-1)//9 # Chai Wah Wu, Jun 28 2025

A376637 The word 1 belongs to the sequence, and whenever a word w belongs to the sequence, then the words consisting of 1's and 2's whose run lengths transform equals w also belong to the sequence.

Original entry on oeis.org

1, 2, 11, 12, 21, 22, 112, 122, 211, 221, 1121, 1122, 1211, 2122, 2211, 2212, 11221, 12112, 12211, 12212, 21121, 21122, 21221, 22112, 112212, 121122, 212211, 221121, 1121122, 1121221, 1122122, 1211221, 1221121, 1221211, 2112122, 2112212, 2122112, 2211211

Offset: 1

Rémy Sigrist, Sep 30 2024

This sequence lists finite smooth words: finite words w composed of 1's and 2's without three or more consecutive equal digits, such that for any k > 0, the k-th iterate of the run lengths transform of w is also a word composed of 1's and 2's without three or more consecutive equal digits.

			The first terms, alongside their run lengths transform, are:
  n   a(n)  RL(a(n))
  --  ----  --------
   1     1         1
   2     2         1
   3    11         2
   4    12        11
   5    21        11
   6    22         2
   7   112        21
   8   122        12
   9   211        12
  10   221        21
  11  1121       211
  12  1122        22
  13  1211       112
  14  2122       112
  15  2211        22
  16  2212       211

Geneviève Paquin, Srĕcko Brlek, Damien Jamet, Extremal generalized smooth words
Rémy Sigrist, Table of n, a(n) for n = 1..10048
Rémy Sigrist, Illustration of the first terms (arrows denotes run lengths transforms)
Rémy Sigrist, PARI program
Index entries for sequences related to Kolakoski sequence

Cf. A000002, A007931, A376638, A376674, A376676, A376685, A376698.

PARI
```
\\ See Links section.
```

A256077 Repeat 2^d times the repunit A002275(d); d = 1, 2, 3...

Original entry on oeis.org

1, 1, 11, 11, 11, 11, 111, 111, 111, 111, 111, 111, 111, 111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 11111, 11111, 11111, 11111, 11111, 11111, 11111, 11111, 11111, 11111, 11111, 11111

Offset: 1

M. F. Hasler, Mar 21 2015

nonn

Yields the length of the n-th (nonempty) binary word (or word over any 2-letter alphabet, like A007931 or A032810 or A032834) in tally mark notation (A000042).

Felix Fröhlich, Table of n, a(n) for n = 1..10000

lim = 5; lst = Table[(10^n - 1)/9, {n, 0, lim}]; Reap@ For[i = 1, i <= lim, i++, Sow@ Table[lst[[i + 1]], {d, 2^i}]] // Flatten // Rest (* Michael De Vlieger, Apr 01 2015 *)

PARI
```
a(n)=10^#binary(n+1)\90
```

def A256077(n): return (10**((n+1).bit_length()-1)-1)//9 # Chai Wah Wu, Nov 07 2024

a(n) = A002275(A000523(n+1)) = A032810(n)-A007931(n) = A032834(n)-A032810(n), etc.

A258410 Nonnegative integers with an equal number of occurrences of all digits in bijective base-2 numeration.

Original entry on oeis.org

4, 5, 18, 20, 21, 24, 25, 27, 70, 74, 76, 77, 82, 84, 85, 88, 89, 91, 98, 100, 101, 104, 105, 107, 112, 113, 115, 119, 270, 278, 282, 284, 285, 294, 298, 300, 301, 306, 308, 309, 312, 313, 315, 326, 330, 332, 333, 338, 340, 341, 344, 345, 347, 354, 356, 357

Offset: 1

Alois P. Heinz, May 29 2015

			4 = 12_bij2, 5 = 21_bij2, 18 = 1122_bij2, 20 = 1212_bij2.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Wikipedia, Bijective numeration

Cf. A007931, A214676, A257869, A258411.

p:= proc(n) local d, m, r; m:= n; r:= 0;
      while m>0 do d:= irem(m, 2, 'm');
        if d=0 then d:=2; m:= m-1 fi;
        r:= r+x^d
      od;
      simplify(r/(x+x^2))::integer
    end:
a:= proc(n) option remember; local k;
      for k from 1+`if`(n=1, 0, a(n-1)) by 1
      while not p(k) do od; k
    end:
seq(a(n), n=1..70);

A284379 Numbers k with digits 3 and 5 only.

Original entry on oeis.org

3, 5, 33, 35, 53, 55, 333, 335, 353, 355, 533, 535, 553, 555, 3333, 3335, 3353, 3355, 3533, 3535, 3553, 3555, 5333, 5335, 5353, 5355, 5533, 5535, 5553, 5555, 33333, 33335, 33353, 33355, 33533, 33535, 33553, 33555, 35333, 35335, 35353, 35355, 35533, 35535

Offset: 1

Jaroslav Krizek, Mar 26 2017

Prime terms are in A020462.

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

Numbers n with digits 5 and k only for k = 0 - 4 and 6 - 9: A169964 (k = 0), A276037 (k = 1), A072961 (k = 2), this sequence (k = 3), A256290 (k = 4), A256291 (k = 6), A284380 (k = 7), A284381 (k = 8), A284382 (k = 9).

[n: n in [1..100000] | Set(IntegerToSequence(n, 10)) subset {3, 5}];

A:= 3,5: B:= [3,5];
for i from 1 to 5 do
  B:= map(t -> (10*t+3,10*t+5), B);
  A:= A, op(B);
od:
A; # Robert Israel, Apr 13 2020

Select[Range[35600], Times @@ Boole@ Map[MemberQ[{3, 5}, #] &, IntegerDigits@ #] > 0 &] (* or *)
Table[FromDigits /@ Union@ Apply[Join, Map[Permutations@ # &, Tuples[{3, 5}, n]]], {n, 5}] // Flatten (* Michael De Vlieger, Mar 27 2017 *)

From Robert Israel, Apr 13 2020: (Start)
a(n) = 2*A007931(n)+A002275(n).
a(2n+1) = 10*a(n)+3.
a(2n+2) = 10*a(n)+5.
G.f. g(x) satisfies g(x) = 10*(x^2+x)*g(x^2) + (3*x+5*x^2)/(1-x^2). (End)

A284633 Numbers n with digits 3 and 6 only.

Original entry on oeis.org

3, 6, 33, 36, 63, 66, 333, 336, 363, 366, 633, 636, 663, 666, 3333, 3336, 3363, 3366, 3633, 3636, 3663, 3666, 6333, 6336, 6363, 6366, 6633, 6636, 6663, 6666, 33333, 33336, 33363, 33366, 33633, 33636, 33663, 33666, 36333, 36336, 36363, 36366, 36633, 36636

Offset: 1

Jaroslav Krizek, Mar 30 2017

All terms after 3 are composite.

Cf. A007931.

Numbers n with digits 6 and k only for k = 0..5 and 7..9: A204093 (k = 0), A284293 (k = 1), A284632 (k = 2), this sequence (k = 3), A284634 (k = 4), A256291 (k = 5), A256292 (k = 7), A284635 (k = 8), A284636 (k = 9).

[n: n in [1..100000] | Set(IntegerToSequence(n, 10)) subset {3, 6}]

Table[Map[FromDigits, Tuples[{3, 6}, {k}]], {k, 5}] // Flatten (* Michael De Vlieger, Mar 30 2017 *)

a(n) = 3*A007931(n). - Michel Marcus, Mar 30 2017

A284920 Numbers with digits 2 and 4 only.

Original entry on oeis.org

2, 4, 22, 24, 42, 44, 222, 224, 242, 244, 422, 424, 442, 444, 2222, 2224, 2242, 2244, 2422, 2424, 2442, 2444, 4222, 4224, 4242, 4244, 4422, 4424, 4442, 4444, 22222, 22224, 22242, 22244, 22422, 22424, 22442, 22444, 24222, 24224, 24242, 24244, 24422, 24424

Offset: 1

Jaroslav Krizek, Apr 05 2017

All terms are even.

Cf. Numbers with digits 2 and k only for k = 0 - 1 and 3 - 9: A169965 (k = 0), A007931 (k = 1), A032810 (k = 3), this sequence (k = 4), A072961 (k = 5), A284632 (k = 6), A284921 (k = 7), A284922 (k = 8), A284923 (k = 9).

[n: n in [1..100000] | Set(IntegerToSequence(n, 10)) subset {2, 4}]

Flatten@ Array[FromDigits /@ Tuples[{2, 4}, #] &, 5] (* Michael De Vlieger, Apr 06 2017 *)

a(n) = 2 * A007931(n).

A291072 Take n-th string over {1,2} in lexicographic order and apply the Watanabe tag system {00, 1011} described in A291067 (but adapted to the alphabet {1,2}) just once.

Original entry on oeis.org

-1, 22, 1, 1, 122, 122, 11, 11, 11, 11, 2122, 2122, 2122, 2122, 111, 211, 111, 211, 111, 211, 111, 211, 12122, 22122, 12122, 22122, 12122, 22122, 12122, 22122, 1111, 1211, 2111, 2211, 1111, 1211, 2111, 2211, 1111, 1211, 2111, 2211, 1111, 1211, 2111, 2211, 112122, 122122, 212122, 222122

Offset: 1

N. J. A. Sloane, Aug 18 2017

sign

Cf. A007931, A284116, A291067, A291068, A291069.

Cf. also A289673, A291073, A291074.

# First define the mapping by defining the strings T1 and T2:
# Work over the alphabet {1,2}
# 11 / 2212 A284116 This is the "Post Tag System"
T1:="11"; T2:="2212";
# 11 / 2122 A291067 These three are from the Watanabe paper
T1:="11"; T2:="2122";
# 11 / 2221 A291068
T1:="11"; T2:="2221";
# 11 / 1222 A291069
T1:="11"; T2:="1222";
with(StringTools):
# the mapping:
f1:=proc(w) local L, ws, w2; global T1,T2;
ws:=convert(w, string);
if ws="-1" then return("-1"); fi;
if ws[1]="1" then w2:=Join([ws, T1], ""); else w2:=Join([ws, T2], "");  fi;
L:=length(w2); if L <= 3 then return("-1"); fi;
w2[4..L]; end;
# Construct list of words over {1,2} (A007931)
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:
WLIST := [seq(a(n), n=1..100)];
# apply the map once:
# this produces A289673, A291072, A291073, A291074
W2:=map(f1,WLIST);

A111066 Numbers with digits 1 and 2 and at least one of each.

Original entry on oeis.org

12, 21, 112, 121, 122, 211, 212, 221, 1112, 1121, 1122, 1211, 1212, 1221, 1222, 2111, 2112, 2121, 2122, 2211, 2212, 2221, 11112, 11121, 11122, 11211, 11212, 11221, 11222, 12111, 12112, 12121, 12122, 12211, 12212, 12221, 12222, 21111, 21112, 21121, 21122, 21211

Offset: 1

Alexandre Wajnberg & Youri Mora, Oct 08 2005

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Equals A007931 minus A000042 and A002276. Supersequence of A214218.

Cf. A043494, A037415, A062289, A000225.

FromDigits /@ Select[ IntegerDigits[ Range[210], 3], Union[ # ] == {1, 2} &] (* Robert G. Wilson v, Oct 09 2005 *)
Union[FromDigits/@Select[Flatten[Table[Tuples[{1,2},n],{n,2,5}],1], Union[#] == {1,2}&]] (* Harvey P. Dale, Sep 05 2013 *)

from itertools import count, islice
def agen():
    for i in count(1):
        s = bin(i+1)[3:].replace('1', '2').replace('0', '1')
        if 0 < s.count('1') < len(s):
            yield int(s)
print(list(islice(agen(), 42))) # Michael S. Branicky, Dec 21 2021

More terms from Robert G. Wilson v, Oct 09 2005

Crossrefs from Charles R Greathouse IV, Aug 03 2010

A213972 List of imprimitive words over the alphabet {1,2}.

Original entry on oeis.org

11, 22, 111, 222, 1111, 1212, 2121, 2222, 11111, 22222, 111111, 112112, 121121, 121212, 122122, 211211, 212121, 212212, 221221, 222222, 1111111, 2222222, 11111111, 11121112, 11211121, 11221122, 12111211, 12121212, 12211221, 12221222, 21112111, 21122112, 21212121

Offset: 1

N. J. A. Sloane, Jun 30 2012

A word w is primitive if it cannot be written as u^k with k>1; otherwise it is imprimitive.
The {0,1} version of this sequence is
00, 11, 000, 111, 0000, 0101, 1010, 1111, 00000, 11111, 000000, 001001, 010010, 010101, 011011, 100100, 101010, 101101, 110110, 111111
but this cannot be included as a sequence in the OEIS since it contains nonzero "numbers" beginning with 0.
This sequence results from A213973 by replacing all digits 3 by 2 and from A213974 by replacing digits 2 by 1 and digits 3 by 2. - M. F. Hasler, Mar 10 2014

A. de Luca and S. Varricchio, Finiteness and Regularity in Semigroups and Formal Languages, Monographs in Theoretical Computer Science, Springer-Verlag, Berlin, 1999. See p. 10.

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

Cf. A213969-A213974.

See A239018 for the analog over the alphabet {1,2,3}.

P:= proc(d) option remember;local m,A;
    A:= map(t -> (10^d-1)/9 + add(10^s, s = t), combinat:-powerset([$0..d-1]));
    for m in numtheory:-divisors(d) minus {d} do
      A:= remove(t -> t = (t mod 10^m)*(10^d-1)/(10^m-1), A);
    od;
    sort(A);
end proc:
IP:= proc(d)
   sort([seq(seq(s*(10^d-1)/(10^m-1), s = P(m)), m=numtheory:-divisors(d) minus {d})]);
end proc:
seq(op(IP(d)), d=1..10); # Robert Israel, Mar 24 2017

j[w_, k_] := FromDigits /@ (Flatten[Table[#, {k}]] & /@ w); Flatten@ Table[ Union@ Flatten[ j[Tuples [{1, 2}, #], n/#] & /@ Most@ Divisors@ n], {n, 9}] (* Giovanni Resta, Mar 24 2017 *)

for(n=1, 10, p=vector(n, i, 10^(n-i))~; forvec(d=vector(n, i, [1, 2]), is_A239017(m=d*p)||print1(m", "))) \\ M. F. Hasler, Mar 10 2014

A213972 = A007931 intersect A239018. - M. F. Hasler, Mar 10 2014

More terms from M. F. Hasler, Mar 10 2014

cp's OEIS Frontend

A284293 Numbers using only digits 1 and 6.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Python

A376637 The word 1 belongs to the sequence, and whenever a word w belongs to the sequence, then the words consisting of 1's and 2's whose run lengths transform equals w also belong to the sequence.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A256077 Repeat 2^d times the repunit A002275(d); d = 1, 2, 3...

Views

Author

Keywords

Comments

Links

Programs

Mathematica

PARI

Python

Formula

A258410 Nonnegative integers with an equal number of occurrences of all digits in bijective base-2 numeration.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

A284379 Numbers k with digits 3 and 5 only.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

A284633 Numbers n with digits 3 and 6 only.

Views

Author

Keywords

Comments

Crossrefs

Programs

Magma

Mathematica

Formula

A284920 Numbers with digits 2 and 4 only.

Views

Author

Keywords

Comments

Crossrefs

Programs

Magma

Mathematica

Formula

A291072 Take n-th string over {1,2} in lexicographic order and apply the Watanabe tag system {00, 1011} described in A291067 (but adapted to the alphabet {1,2}) just once.

Views

Author

Keywords

Crossrefs

Programs