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A004022 Primes of form (10^n - 1)/9.
(Formerly M4816)
11, 1111111111111111111, 11111111111111111111111 (list; graph; refs; listen; history; text; internal format)



The next term corresponds to n = 317 and is too large to include: see A004023, A046413.

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^n-1)/9 is n, if n = pk then (10^pk-1)=(10^p)^k-1 => (10^p-1)/9 = q and q divides (10^n-1). This follows from the identity a^n-b^n=(a-b)(a^(n-1)+a^(n-2)b+...+b^n-1). - Cino Hilliard, Dec 23 2008


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.

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).

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.


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.

Makoto Kamada, Factorizations of 11...11 (Repunit)

Andy Steward, Prime Generalized Repunits

S. S. Wagstaff, Jr., The Cunningham Project


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


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 *)


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

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


See A004023 for the number of 1's. Cf. A046413.

Sequence in context: A066953 A213645 * A243534 A241570 A083344 A063863

Adjacent sequences:  A004019 A004020 A004021 * A004023 A004024 A004025




N. J. A. Sloane.


Edited by Max Alekseyev, Nov 15 2010



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Last modified December 20 02:13 EST 2014. Contains 252240 sequences.