A032917 -id:A032917 - cp's OEIS frontend

A020451 Primes that contain digits 1 and 3 only.

3, 11, 13, 31, 113, 131, 311, 313, 331, 3313, 3331, 11113, 11131, 11311, 13313, 13331, 31333, 33113, 33311, 33331, 113111, 113131, 131111, 131113, 131311, 311111, 313133, 313331, 313333, 331333, 333131, 333331, 1111333, 1131113, 1131131, 1131133, 1131331

Offset: 1

David W. Wilson

Jason Bard, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)

Subsequence of A030096, A045429, and A032917.

[p: p in PrimesUpTo(1131331) | Set(Intseq(p)) subset [1,3]]; // Bruno Berselli, Jul 27 2012

N:= 8: # to get all a(n) with at most N digits
S:= {}:
for d from 1 to N do
  r:= (10^d-1)/9;
  S:= S union select(isprime,map(`+`,map(convert,combinat[powerset]
      ({seq(2*10^i,i=0..d-1)}),`+`),r));
od:
S; # if using Maple 11 or earlier, uncomment the next line
# sort(convert(S,list)); # Robert Israel, May 04 2015

Flatten[Table[Select[FromDigits/@Tuples[{1,3},n],PrimeQ],{n,7}]] (* Vincenzo Librandi, Jul 27 2012 *)

from sympy import primerange
def checkd(a, c):
    b =  set(int(i) for i in set(str(a)))
    return b.issubset(c)
for n in primerange(2, 2000000):
    if checkd(n, [1, 3]):
        print(n)
# Abhiram R Devesh, May 04 2015

A284293 Numbers using only digits 1 and 6.

Original entry on oeis.org

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

A284632 Numbers n with digits 2 and 6 only.

Original entry on oeis.org

2, 6, 22, 26, 62, 66, 222, 226, 262, 266, 622, 626, 662, 666, 2222, 2226, 2262, 2266, 2622, 2626, 2662, 2666, 6222, 6226, 6262, 6266, 6622, 6626, 6662, 6666, 22222, 22226, 22262, 22266, 22622, 22626, 22662, 22666, 26222, 26226, 26262, 26266, 26622, 26626

Offset: 1

Jaroslav Krizek, Mar 30 2017

All terms after 2 are composite.

Cf. A032917.

Numbers n with digits 6 and k only for k = 0..5 and 7..9: A204093 (k = 0), A284293 (k = 1), this sequence (k = 2), A284633 (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 {2, 6}]

Table[Map[FromDigits, Tuples[{2, 6}, {k}]], {k, 5}] // Flatten (* Michael De Vlieger, Mar 30 2017 *)

a(n) = 2 * A032917(n).

A213973 List of imprimitive words over the alphabet {1,3}.

Original entry on oeis.org

11, 33, 111, 333, 1111, 1313, 3131, 3333, 11111, 33333, 111111, 113113, 131131, 131313, 133133, 311311, 313131, 313313, 331331, 333333, 1111111, 3333333, 11111111, 11131113, 11311131, 11331133, 13111311, 13131313, 13311331, 13331333, 31113111, 31133113, 31313131

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 A213972 by replacing all digits 2 by 3, and from A213974 by replacing all digits 2 by 1. - 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.

Cf. A213969-A213974.

for(n=1, 8, p=2*vector(n, i, 10^(n-i))~; forvec(d=vector(n, i, [1, 3]/2), is_A239017(m=d*p)||print1(m", ")))

A213973 = A032917 intersect A239018. - M. F. Hasler, Mar 10 2014

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

A239019 Numbers which are not primitive words over the alphabet {0,...,9} (when written in base 10).

Original entry on oeis.org

11, 22, 33, 44, 55, 66, 77, 88, 99, 111, 222, 333, 444, 555, 666, 777, 888, 999, 1010, 1111, 1212, 1313, 1414, 1515, 1616, 1717, 1818, 1919, 2020, 2121, 2222, 2323, 2424, 2525, 2626, 2727, 2828, 2929, 3030, 3131, 3232, 3333, 3434, 3535, 3636, 3737, 3838, 3939, 4040, 4141, 4242, 4343, 4444, 4545, 4646, 4747, 4848, 4949

Offset: 1

M. F. Hasler, Mar 08 2014

A word is primitive iff it is not a power, i.e., repetition, of a subword. The only non-primitive words with a prime number of letters (here: digits) are the repdigit numbers. Thus, the first nontrivial terms of this sequence are 1010,1212,...
This sequence does *not* contain all non-primitive words over the alphabet {0,...,9}, namely, it excludes those which would be numbers with leading zeros: 00,000,0000,0101,0202,...
Lists of non-primitive words over a sub-alphabet of {1...9}, like A213972, A213973, A213974, A239018, ... are given as intersection of this with the set of all words in that alphabet, e.g., A007931, A032810, A032917, A007932, ...

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

F:= proc(d) local p,R,q;
  R:= {seq(x*(10^d-1)/9, x=1..9)};
  for p in numtheory:-factorset(d) minus {d} do
    q:= d/p;
    R:= R union {seq(x*(10^d-1)/(10^q-1),x=10^(q-1)..10^q-1)};
  od:
  sort(convert(R,list))
end proc:
[seq(op(F(i)),i=2..4)]; # Robert Israel, Nov 14 2017

is_A239019(n)=fordiv(#n=digits(n),L,L<#n && n==concat(Col(vector(#n/L,i,1)~*vecextract(n,2^L-1))~)&&return(1))

A213084 Numbers consisting of ones and eights.

Original entry on oeis.org

1, 8, 11, 18, 81, 88, 111, 118, 181, 188, 811, 818, 881, 888, 1111, 1118, 1181, 1188, 1811, 1818, 1881, 1888, 8111, 8118, 8181, 8188, 8811, 8818, 8881, 8888, 11111, 11118, 11181, 11188, 11811, 11818, 11881, 11888, 18111, 18118, 18181, 18188, 18811, 18818

Offset: 1

Jens Ahlström, Jun 05 2012

One and eight begin with vowels. The subsequence of primes begins 11, 181, 811, 1181, 1811, 8111. - Jonathan Vos Post, Jun 14 2012

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

Cf. A020456 (primes in this sequence).

Cf. numbers consisting of 1s and ks: A007088 (k=0), A007931 (k=2), A032917 (k=3), A032822 (k=4), A276037 (k=5), A284293 (k=6), A276039 (k=7), A284294 (k=9).

Flatten[Table[FromDigits/@Tuples[{1,8},n],{n,5}]] (* Harvey P. Dale, Aug 27 2014 *)

is(n) = #setintersect(vecsort(digits(n), , 8), [0, 2, 3, 4, 5, 6, 7, 9])==0 \\ Felix Fröhlich, Sep 09 2019

res = []
i = 0
while len (res) < 260:
    for c in str(i):
        if c in '18':
            continue
        else:
            break
    else:
        res.append(i)
    i = i + 1
print(res)

def a(n): return int(bin(n+1)[3:].replace('1', '8').replace('0', '1'))
print([a(n) for n in range(1, 45)]) # Michael S. Branicky, Jun 26 2025

A284294 Numbers using only digits 1 and 9.

Original entry on oeis.org

1, 9, 11, 19, 91, 99, 111, 119, 191, 199, 911, 919, 991, 999, 1111, 1119, 1191, 1199, 1911, 1919, 1991, 1999, 9111, 9119, 9191, 9199, 9911, 9919, 9991, 9999, 11111, 11119, 11191, 11199, 11911, 11919, 11991, 11999, 19111, 19119, 19191, 19199, 19911, 19919

Offset: 1

Jaroslav Krizek, Mar 25 2017

Product of digits of terms is a power of 9; subsequence of A284295.

Prime terms are in A020457.

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), A284293 (k = 6), A276039 (k = 7), A213084 (k = 8), this sequence (k = 9).

[n: n in [1..20000] | Set(IntegerToSequence(n, 10)) subset {1, 9}];

Join @@ (FromDigits /@ Tuples[{1,9}, #] & /@ Range[5]) (* Giovanni Resta, Mar 25 2017 *)

The sum of first 2^n terms is (5*20^n + 38*10^n - 95*2^n + 1420)/171. - Giovanni Resta, Mar 25 2017

A032913 Numbers whose set of base-6 digits is {1,3}.

Original entry on oeis.org

1, 3, 7, 9, 19, 21, 43, 45, 55, 57, 115, 117, 127, 129, 259, 261, 271, 273, 331, 333, 343, 345, 691, 693, 703, 705, 763, 765, 775, 777, 1555, 1557, 1567, 1569, 1627, 1629, 1639, 1641, 1987, 1989, 1999, 2001, 2059, 2061, 2071, 2073

Offset: 1

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A007092, A032917.

[n: n in [1..2500] | Set(IntegerToSequence(n, 6)) subset {1, 3}]; // Vincenzo Librandi, Jun 01 2012

Flatten[Table[FromDigits[#,6]&/@Tuples[{1,3},n],{n,5}]] (* Harvey P. Dale, Nov 16 2011 *)

def A032913(n): return (int(bin(m:=n+1)[3:],6)<<1) + (6**(m.bit_length()-1)-1)//5 # Chai Wah Wu, Oct 13 2023

A032914 Numbers whose set of base-7 digits is {1,3}.

Original entry on oeis.org

1, 3, 8, 10, 22, 24, 57, 59, 71, 73, 155, 157, 169, 171, 400, 402, 414, 416, 498, 500, 512, 514, 1086, 1088, 1100, 1102, 1184, 1186, 1198, 1200, 2801, 2803, 2815, 2817, 2899, 2901, 2913, 2915, 3487, 3489, 3501, 3503, 3585, 3587

Offset: 1

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A007093, A032917.

[n: n in [1..4000] | Set(IntegerToSequence(n, 7)) subset {1, 3}]; // Vincenzo Librandi, Jun 01 2012

Flatten[Table[FromDigits[#,7]&/@Tuples[{1,3},n],{n,5}]] (* Vincenzo Librandi, Jun 01 2012 *)

def A032914(n): return (int(bin(m:=n+1)[3:],7)<<1) + (7**(m.bit_length()-1)-1)//6 # Chai Wah Wu, Oct 13 2023

A032915 Numbers whose set of base-8 digits is {1,3}.

Original entry on oeis.org

1, 3, 9, 11, 25, 27, 73, 75, 89, 91, 201, 203, 217, 219, 585, 587, 601, 603, 713, 715, 729, 731, 1609, 1611, 1625, 1627, 1737, 1739, 1753, 1755, 4681, 4683, 4697, 4699, 4809, 4811, 4825, 4827, 5705, 5707, 5721, 5723, 5833, 5835

Offset: 1

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A007094, A032917.

[n: n in [1..6000] | Set(IntegerToSequence(n, 8)) subset {1, 3}]; // Vincenzo Librandi, Jun 01 2012

Flatten[Table[FromDigits[#,8]&/@Tuples[{1,3},n],{n,5}]] (* Vincenzo Librandi, Jun 01 2012 *)

def A032915(n): return (int(bin(m:=n+1)[3:],8)<<1) + ((1<<3*(m.bit_length()-1))-1)//7 # Chai Wah Wu, Oct 13 2023

cp's OEIS Frontend

A020451 Primes that contain digits 1 and 3 only.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Python

A284293 Numbers using only digits 1 and 6.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Python

A284632 Numbers n with digits 2 and 6 only.

Views

Author

Keywords

Comments

Crossrefs

Programs

Magma

Mathematica

Formula

A213973 List of imprimitive words over the alphabet {1,3}.

Views

Author

Keywords

Comments

References

Crossrefs

Programs

PARI

Formula

Extensions

A239019 Numbers which are not primitive words over the alphabet {0,...,9} (when written in base 10).

Views

Author

Keywords

Comments

Links

Programs

Maple

PARI

A213084 Numbers consisting of ones and eights.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Python

A284294 Numbers using only digits 1 and 9.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A032913 Numbers whose set of base-6 digits is {1,3}.

Views

Author

Keywords