A116692 -id:A116692 - cp's OEIS frontend

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

11, 1111111111111111111, 11111111111111111111111

Offset: 1

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

A055387 2, 3, 5, 7, together with primes such that there is a nontrivial rearrangement of the digits which is a prime.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 31, 37, 71, 73, 79, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 239, 241, 251, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 359, 367, 373, 379, 389, 397, 401, 419, 421

Offset: 1

Asher Auel, May 05 2000

Union of {2, 3, 5, 7} and A225035.

Cf. A004022, A007933, A007935, A052902, A116692, A225035, A359136-A359139.

Edited by N. J. A. Sloane, Jan 22 2023

A030291 Primes with at most two different digits.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 113, 131, 151, 181, 191, 199, 211, 223, 227, 229, 233, 277, 311, 313, 331, 337, 353, 373, 383, 433, 443, 449, 499, 557, 577, 599, 661, 677, 727, 733

Offset: 1

Patrick De Geest

The one-digit primes (2, 3, 5, 7) followed by the union of A004022 and A235154. - Jeppe Stig Nielsen, Feb 17 2021

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..21936 (first 1000 terms from T. D. Noe)

Cf. A004022, A032758, A116692, A235154.

Select[Range[1000], PrimeQ[#] && Length[Union[RealDigits[#][[1]]]] <= 2 &]
Select[Prime[Range[200]],Count[DigitCount[#],0]>7&] (* Harvey P. Dale, Jul 14 2017 *)

Offset corrected by Arkadiusz Wesolowski, Sep 13 2011

A069598 Primes using only one nonzero digit (with zero digits allowed).

Original entry on oeis.org

2, 3, 5, 7, 11, 101, 10111, 101111, 1011001, 1100101, 10010101, 10011101, 10100011, 10101101, 10110011, 10111001, 11000111, 11100101, 11110111, 11111101, 100100111, 100111001, 101001001, 101001011, 101100011, 101101111

Offset: 1

Amarnath Murthy, Mar 25 2002

From the fifth term onwards only zeros and ones are used.
Any number composed only of a string of zeros and d's where d is a digit > 1 is divisible by d and not prime, so this sequence is the single-digit primes {2,3,5,7} UNION A020449.
The repunit primes (A004022) and A116692 are subsequences.

Cf. A020449, A004022, A116692.

Edited by Rick L. Shepherd, Sep 04 2009

A088562 Palindromic primes using at most two distinct digits.

Original entry on oeis.org

2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 11311, 11411, 13331, 15551, 16661, 18181, 19991, 32323, 33533, 35353, 72227, 72727, 74747, 75557, 76667, 77377, 77477, 77977, 78787, 78887, 79997, 94949, 95959

Offset: 1

Amarnath Murthy, Nov 28 2003

Union of A056730 and A116692. - Arkadiusz Wesolowski, Sep 13 2011

Jon E. Schoenfield, Table of n, a(n) for n = 1..12320 (all terms < 10^25)

Cf. A056730, A116692.

Select[Prime[Range[10000]],PalindromeQ[#]&&Count[DigitCount[#],0]>7&] (* Harvey P. Dale, Jul 31 2023 *)

More terms from Arkadiusz Wesolowski, Sep 13 2011

A182183 Numbers k such that the divisors of k are divisible by all digits of their divisors.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 22, 24, 33, 44, 55, 66, 77, 88, 99, 132, 264, 1111111111111111111, 2222222222222222222, 3333333333333333333, 4444444444444444444, 5555555555555555555, 6666666666666666666, 7777777777777777777, 8888888888888888888

Offset: 1

Jaroslav Krizek, Apr 17 2012

Subsequence of A209933 (numbers that are divisible by all digits of their divisors).
All divisors of numbers in this sequence are also in the sequence.
The primitive elements of this sequence are A116692. No member of this sequence is divisible by a prime outside this sequence. - Charles R Greathouse IV, Apr 17 2012

			Number 48 with divisors 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 is not in the sequence because 6 is not a divisor of 16.

Subsequence of A034838. A116692 is a subsequence.
Cf. A209933.
Cf. A027751.
Cf. A027751.

import Data.List ((\\))
a182183 n = a182183_list !! (n-1)
a182183_list = f a209933_list [1] where
   f (x:xs) ys =
     if null (a027751_row x \\ ys) then x : f xs (x : ys) else f xs ys
-- Reinhard Zumkeller, Apr 19 2012

all(n)=my(v=vecsort(eval(Vec(Str(n))),,8)); if(v[1]==0, return(0)); for(i=1,#v,if(n%v[i],return(0)));1
is(n)=fordiv(n,d,if(!all(d),return(0)));1 \\ Charles R Greathouse IV, Apr 17 2012

a(24)-a(31) from Charles R Greathouse IV, Apr 17 2012

cp's OEIS Frontend

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

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A055387 2, 3, 5, 7, together with primes such that there is a nontrivial rearrangement of the digits which is a prime.

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A030291 Primes with at most two different digits.

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A069598 Primes using only one nonzero digit (with zero digits allowed).

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A088562 Palindromic primes using at most two distinct digits.

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A182183 Numbers k such that the divisors of k are divisible by all digits of their divisors.

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