

A004023


Indices of prime repunits: numbers n such that 11...111 = (10^n  1)/9 is prime.
(Formerly M2114)


107




OFFSET

1,1


COMMENTS

People who search for repunit primes or repdigit primes may be looking for this entry.
The indices of primes with digital product (i.e., product of digits) equal to 1.
As of August, 2014, only the first five repunits, through (10^10311)/9, have been proved prime. The last four repunits are known only to be probable primes and have not been proved to be prime.  Robert Baillie, Aug 17 2014
These indices p must also be prime. If p is not prime, say p = mn, then 10^mn  1 =((10^m)^n)  1 => 10^m  1 divides 10^mn  1. Since 9 divides 10^m  1 or (10^m1)/9 = q, it follows q divides (10^p1)/9. This is a result of the identity, a^n  b^n = (ab)(a^(n1) + a^(n2)b + ... + b^(n1).  Cino Hilliard, Dec 23 2008
The numbers R_n = 11...111 = (10^n  1)/9 with n in this sequence A004023, except for n = 2, are prime repunits in base ten, so they are prime Brazilian numbers belonging to A085104. [See Links: Les nombres brésiliens.]  Bernard Schott, Dec 24 2012
Primality confirmed for (10^(49081)1)/9 using sieve methods in Mathematica .  Shivam N. Patel, Dec 17 2015


REFERENCES

J. Brillhart et al., Factorizations of b^n + 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
J.M. De Koninck, Ces nombres qui nous fascinent, Entry 19, pp 6, Ellipses, Paris 2008.
Graham, Knuth and Patashnik, Concrete Mathematics, AddisonWesley, 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).


LINKS

Table of n, a(n) for n=1..9.
J. Brillhart, Letter to N. J. A. Sloane, Aug 08 1970
J. Brillhart et al., Factorizations of b^n + 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
C. K. Caldwell, The Prime Glossary, Repunit
Patrick De Geest, Circular Primes
Giovanni Di Maria, Repunit Primes Project
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927930. [Annotated scanned copy]
Harvey Dubner, New probable prime repunit, R(49081), Number Theory List, Sep 09 1999.
Harvey Dubner, Repunit R49081 is a probable prime, Math. Comp., 71 (2001), 833835.
H. Dubner, Posting to Number Theory List : Apr 03 2007
Martianus Frederic Ezerman, Bertrand Meyer, and Patrick Solé, On Polynomial Pairs of Integers, arXiv:1210.7593 [math.NT], 20122014.
Martianus Frederic Ezerman, Bertrand Meyer and Patrick Solé, On Polynomial Pairs of Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.3.5.
Makoto Kamada, Factorizations of 11...11 (Repunit).
H. Lifchitz, Mersenne and Fermat primes field
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avriljuin 2010, pages 3038; included here with permission from the editors of Quadrature.
Andy Steward, Prime Generalized Repunits
S. S. Wagstaff, Jr., The Cunningham Project
E. Wegrzynowski, Nombres 1_[n] premiers
Eric Weisstein's World of Mathematics, Repunit
Eric Weisstein's World of Mathematics, Integer Sequence Primes
H. C. Williams and Harvey Dubner, The primality of R1031, Math. Comp., 47(176), Oct 1986, 703711.
Index entries for primes involving repunits


EXAMPLE

2 appears because the 2digit repunit 11 is prime.
19 appears because the 19digit repunit 1111111111111111111 is prime.


MATHEMATICA

Select[Range[271000], PrimeQ[FromDigits[PadRight[{}, #, 1]]] &] (* Harvey P. Dale, Nov 05 2011 *)
repUnsUpTo[k_]:= ParallelMap[If[PrimeQ[#] && PrimeQ[(10^#  1)/9], #, Nothing] &,
Range[k]]; repUnsUpTo[5000] (* Mikk Heidemaa, Apr 24 2017 *)


PROG

(PARI) forprime(x=2, 20000, if(ispseudoprime((10^x1)/9), print1(x", "))) \\ Cino Hilliard, Dec 23 2008
(MAGMA) [p: p in PrimesUpTo(500)  IsPrime((10^p  1) div 9)]; // Vincenzo Librandi, Nov 06 2014


CROSSREFS

See A004022 for the actual primes.
Cf. A055557, A002275, A085104.
Sequence in context: A105907 A018696 A175451 * A186682 A031030 A083689
Adjacent sequences: A004020 A004021 A004022 * A004024 A004025 A004026


KEYWORD

hard,nonn,nice,more


AUTHOR

N. J. A. Sloane


EXTENSIONS

49081 found by Harvey Dubner  posting to Number Theory List (NMBRTHRY(AT)LISTSERV.NODAK.EDU) Sep 09, 1999.
86453 found using pfgw (a faster version of PrimeForm) on Oct 26 2000 by Lew Baxter (ldenverb(AT)hotmail.com)  posting to Number Theory List (NMBRTHRY(AT)LISTSERV.NODAK.EDU), Oct 26, 2000.
a(8) = 109297 was apparently discovered independently by (in alphabetical order) Paul Bourdelais and Harvey Dubner (harvey(AT)dubner.com) around Mar 2628 2007.
A new probable prime repunit, R(270343), was found Jul 11 2007 by Maksym Voznyy (mvoznyy0526(AT)ROGERS.COM) and Anton Budnyy.
R(270343) subsequently confirmed as a(9) (see Repunit Primes Project link) by Robert Price, Dec 14 2010
Link to Repunit Primes Project site updated by Felix Fröhlich, Oct 19 2014


STATUS

approved



