A020457 -id:A020457 - cp's OEIS frontend

A385776 Primes having only {1, 2, 9} as digits.

2, 11, 19, 29, 191, 199, 211, 229, 911, 919, 929, 991, 1129, 1229, 1291, 1999, 2111, 2129, 2221, 2999, 9199, 9221, 9929, 11119, 11299, 12119, 12211, 12911, 12919, 19121, 19211, 19219, 19919, 19991, 21121, 21191, 21211, 21221, 21911, 21929, 21991

Offset: 1

Jason Bard, Jul 09 2025

Supersequence of A020450, A020457, A020460.

Cf. A000040.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [1, 2, 9]];

Flatten[Table[Select[FromDigits /@ Tuples[{1, 2, 9}, n], PrimeQ], {n, 7}]]

primes_with(n=50, show=0, L=[1, 2, 9])={for(d=1, 1e9, my(t, u=vector(d, i, 10^(d-i))~); forvec(v=vector(d, i, [1+!(L[1]||(i>1&&i

from gmpy2 import is_prime
from itertools import count, islice, product
def primes_with(digits):  # generator of primes having only set(digits) as digits
    S, E = "".join(sorted(set(digits) - {'0'})), "".join(sorted(set(digits) & set("1379")))
    yield from (p for p in [2, 3, 5, 7] if str(p) in digits)
    yield from (t for d in count(2) for s in S for m in product(digits, repeat=d-2) for e in E if is_prime(t:=int(s+"".join(m)+e)))
print(list(islice(primes_with("129"), 41))) # Michael S. Branicky, Jul 11 2025

A004022 Primes of the form (10^k - 1)/9. Also called repunit primes or repdigit primes.

Original entry on oeis.org

11, 1111111111111111111, 11111111111111111111111

Offset: 1

N. J. A. Sloane

The next term corresponds to k = 317 and is too large to include: see A004023.
Also called repunit primes or prime repunits.
Also, primes with digital product = 1.
The number of 1's in these repunits must also be prime. Since the number of 1's in (10^k-1)/9 is k, if k = p*m then (10^(p*m)-1) = (10^p)^m-1 => (10^p-1)/9 = q and q divides (10^k-1). This follows from the identity a^k - b^k = (a-b)*(a^(k-1) + a^(k-2)*b + ... + b^(k-1)). - Cino Hilliard, Dec 23 2008
A subset of A020449, ..., A020457, A036953, ..., cf. link to OEIS index. - M. F. Hasler, Jul 27 2015
The terms in this sequence, except 11 which is not Brazilian, are prime repunits in base ten, so they are Brazilian primes belonging to A085104 and A285017. - Bernard Schott, Apr 08 2017

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, p. 11. Graham, Knuth and Patashnik, Concrete mathematics, Addison-Wesley, 1994; see p. 146, problem 22.
M. Barsanti, R. Dvornicich, M. Forti, T. Franzoni, M. Gobbino, S. Mortola, L. Pernazza and R. Romito, Il Fibonacci N. 8 (included in Il Fibonacci, Unione Matematica Italiana, 2011), 2004, Problem 8.10.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 60.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 114.

T. D. Noe, Table of n, a(n) for n = 1..5
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Dmytro S. Inosov and Emil Vlasák, Cryptarithmically unique terms in integer sequences, arXiv:2410.21427 [math.NT], 2024. See p. 18.
Makoto Kamada, Factorizations of 11...11 (Repunit).
D. H. Lehmer, On the number (10^23-1)/9, Bull. Amer. Math. Soc. 35 (1929), 349-350.
James Maynard and Brady Haran, Primes without a 7, Numberphile video (2019)
Andy Steward, Prime Generalized Repunits
S. S. Wagstaff, Jr., The Cunningham Project
Index to entries for primes with digits in a given set.

Subsequence of A020449.
A116692 is another version of repunit primes or repdigit primes. - N. J. A. Sloane, Jan 22 2023
See A004023 for the number of 1's.
Cf. A046413.

[a: n in [0..300] | IsPrime(a) where a is (10^n - 1) div 9 ]; // Vincenzo Librandi, Nov 08 2014

lst={}; Do[If[PrimeQ[p = (10^n - 1)/9], AppendTo[lst, p]], {n, 10^2}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 22 2008 *)
Select[Table[(10^n - 1) / 9, {n, 500}], PrimeQ] (* Vincenzo Librandi, Nov 08 2014 *)
Select[Table[FromDigits[PadRight[{},n,1]],{n,30}],PrimeQ] (* Harvey P. Dale, Apr 07 2018 *)

forprime(x=2,20000,if(ispseudoprime((10^x-1)/9),print1((10^x-1)/9","))) \\ Cino Hilliard, Dec 23 2008

from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
    yield from (t for t in (int("1"*k) for k in count(1)) if isprime(t))
print(list(islice(agen(), 4))) # Michael S. Branicky, Jun 09 2022

a(n) = A002275(A004023(n)).

Edited by Max Alekseyev, Nov 15 2010

Name expanded by N. J. A. Sloane, Jan 22 2023

A260271 Primes having only {1, 4, 9} as digits.

Original entry on oeis.org

11, 19, 41, 149, 191, 199, 419, 449, 491, 499, 911, 919, 941, 991, 1499, 1949, 1999, 4111, 4441, 4919, 4999, 9199, 9419, 9491, 9941, 9949, 11119, 11149, 11411, 11491, 11941, 14149, 14411, 14419, 14449, 19141, 19441, 19919, 19949, 19991, 41141, 41149, 41411

Offset: 1

Vincenzo Librandi, Jul 23 2015

A020452, A020457 and A020466 are subsequences.

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

Cf. similar sequences listed in A260266.

Cf. A020452, A020457, A020466.

[p: p in PrimesUpTo(5*10^4) | Set(Intseq(p)) subset [1, 4, 9]];

Select[Prime[Range[5 10^3]], Complement[IntegerDigits[#], {1, 4, 9}]=={} &]

A260893 Primes having only {1, 7, 9} as digits.

Original entry on oeis.org

7, 11, 17, 19, 71, 79, 97, 179, 191, 197, 199, 719, 797, 911, 919, 971, 977, 991, 997, 1117, 1171, 1777, 1979, 1997, 1999, 7177, 7717, 7919, 9199, 9719, 9791, 11117, 11119, 11171, 11177, 11197, 11717, 11719, 11777, 11779, 11971, 17117, 17191, 17791, 17911

Offset: 1

Vincenzo Librandi, Aug 11 2015

A020455, A020457 and A020471 are subsequences.

Subsequence of A030096.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Index to entries for primes with digits in a given set

Cf. similar sequences listed in A260889.

Cf. A000040, A020455, A020457, A020471, A030096.

[p: p in PrimesUpTo(3*10^4) | Set(Intseq(p)) subset [1, 7, 9]];

Select[Prime[Range[3 10^3]], Complement[IntegerDigits[#], {1, 7, 9}] == {}&]

A329761 Primes having only {1, 3, 9} as digits.

Original entry on oeis.org

3, 11, 13, 19, 31, 113, 131, 139, 191, 193, 199, 311, 313, 331, 911, 919, 991, 1193, 1319, 1399, 1913, 1931, 1933, 1993, 1999, 3119, 3191, 3313, 3319, 3331, 3391, 3911, 3919, 3931, 9133, 9199, 9311, 9319, 9391, 9931, 11113, 11119, 11131, 11311, 11393, 11399

Offset: 1

Alois P. Heinz, Nov 20 2019

Original name was: Primes whose product of decimal digits is a power of 3.

Primes whose digit set is a subset of {1,3,9}.

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Marianne Freiberger, Primes without 7s.
James Maynard, Primes with restricted digits, arXiv:1604.01041 [math.NT], 2016.
James Maynard and Brady Haran, Primes without a 7, Numberphile video (2019)
Index to entries for primes with digits in a given set

Subsequence of A030096.

Cf. A000040, A000244, A007954, A020457, A174813.

[p: p in PrimesUpTo(12000) | Set(Intseq(p)) subset [1,3,9]]; // Vincenzo Librandi, Jan 02 2019

Select[Prime[Range[1500]],IntegerQ[Log[3,Times@@IntegerDigits[#]]]&] (* or *) Table[Select[FromDigits/@Tuples[{1,3,9},n],PrimeQ],{n,5}]// Flatten (* Harvey P. Dale, Dec 31 2019 *)

{ A000040 } intersect { A174813 }.

a(n) in { A000040 } and A007954(a(n)) in { A000244 }.

Name changed by Sean A. Irvine, Jul 20 2025

A363023 Primes having only {1, 6, 9} as digits.

Original entry on oeis.org

11, 19, 61, 191, 199, 619, 661, 691, 911, 919, 991, 1619, 1669, 1699, 1999, 6199, 6619, 6661, 6691, 6911, 6961, 6991, 9161, 9199, 9619, 9661, 11119, 11161, 11699, 11969, 16111, 16619, 16661, 16691, 16699, 19661, 19699, 19919, 19961, 19991, 61169, 61961

Offset: 1

Harvey P. Dale, May 13 2023

Jason Bard, Table of n, a(n) for n = 1..10000 (first 2500 terms from Harvey P. Dale)
Index to entries for primes with digits in a given set

Cf. A020454 (1 and 6), A020457 (1 and 9).

Cf. A385776.

Table[Select[Flatten[10#+{1,9}&/@FromDigits/@Tuples[{1,6,9},n]],PrimeQ],{n,4}]//Flatten

A385781 Primes having only {1, 5, 9} as digits.

Original entry on oeis.org

5, 11, 19, 59, 151, 191, 199, 599, 911, 919, 991, 1151, 1511, 1559, 1951, 1999, 5119, 5519, 5591, 9151, 9199, 9511, 9551, 11119, 11159, 11519, 11551, 11959, 15199, 15511, 15551, 15559, 15919, 15959, 15991, 19559, 19919, 19991, 51151, 51199, 51511, 51551

Offset: 1

Jason Bard, Jul 13 2025

Supersequence of A020453, A020457, A020468.

Cf. A000040, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [1, 5, 9]];

Flatten[Table[Select[FromDigits /@ Tuples[{1, 5, 9}, n], PrimeQ], {n, 7}]]

primes_with(, 1, [1, 5, 9]) \\ uses function in A385776

print(list(islice(primes_with("159"), 41))) # uses function/imports in A385776

A385783 Primes having only {1, 8, 9} as digits.

Original entry on oeis.org

11, 19, 89, 181, 191, 199, 811, 881, 911, 919, 991, 1181, 1811, 1889, 1999, 8111, 8191, 8819, 8999, 9181, 9199, 9811, 11119, 11981, 18119, 18181, 18191, 18199, 18899, 18911, 18919, 19181, 19819, 19889, 19891, 19919, 19991, 81119, 81181, 81199, 81899, 81919

Offset: 1

Jason Bard, Jul 13 2025

Supersequence of A020456, A020457, A020472.

Cf. A000040, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [1, 8, 9]];

Flatten[Table[Select[FromDigits /@ Tuples[{1, 8, 9}, n], PrimeQ], {n, 7}]]

primes_with(, 1, [1, 8, 9]) \\ uses function in A385776

print(list(islice(primes_with("189"), 41))) # uses function/imports in A385776

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

A036309 Composite numbers whose prime factors contain no digits other than 1 and 9.

Original entry on oeis.org

121, 209, 361, 1331, 2101, 2189, 2299, 3629, 3781, 3971, 6859, 10021, 10109, 10901, 14641, 17309, 17461, 18829, 21989, 23111, 24079, 25289, 36481, 37981, 38009, 39601, 39919, 41591, 43681, 68951, 71839, 75449, 101189, 110231, 111199, 119911

Offset: 1

Patrick De Geest, Dec 15 1998

All terms are a product of at least two terms of A020457. - David A. Corneth, Oct 09 2020

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 360 terms from Harvey P. Dale)
Index entries for sequences related to prime factors.

Cf. A020457, A036302-A036325.

Select[Range[120000],CompositeQ[#]&&SubsetQ[{1,9},Union[Flatten[ IntegerDigits /@ FactorInteger[ #][[All,1]]]]]&] (* Harvey P. Dale, Mar 30 2019 *)

Sum_{n>=1} 1/a(n) = Product_{p in A020457} (p/(p - 1)) - Sum_{p in A020457} 1/p - 1 = 0.0200389643... . - Amiram Eldar, May 18 2022

cp's OEIS Frontend

A385776 Primes having only {1, 2, 9} as digits.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

A004022 Primes of the form (10^k - 1)/9. Also called repunit primes or repdigit primes.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

Formula

Extensions

A260271 Primes having only {1, 4, 9} as digits.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

A260893 Primes having only {1, 7, 9} as digits.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

A329761 Primes having only {1, 3, 9} as digits.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A363023 Primes having only {1, 6, 9} as digits.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A385781 Primes having only {1, 5, 9} as digits.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

A385783 Primes having only {1, 8, 9} as digits.

Views