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

11, 1111111111111111111, 11111111111111111111111

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• bref

Comments

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

References

• 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.

Links

• 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.

Crossrefs

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.

Programs

Magma

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

Mathematica

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 *)

PARI

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

Python

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

Formula

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

Extensions

Edited by Max Alekseyev, Nov 15 2010

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