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A004023
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Indices of prime repunits: numbers n such that 11...111 = (10^n - 1)/9 is prime.
(Formerly M2114)
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94
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OFFSET
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1,1
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COMMENTS
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The indices of primes with digital product (i.e. product of digits) equal to 1.
The larger terms may only correspond to probable primes.
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^m-1)/9 = q, it follows q divides (10^p-1)/9. This is a result of the identity, a^n-b^n = (a-b)(a^(n-1) + a^(n-2)b + ... + b^(n-1). [Cino Hilliard (hillcino368(AT)hotmail.com), Dec 23 2008]
The numbers R_n = 11...111 = (10^n - 1)/9 with n in this sequence A004023, 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]
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REFERENCES
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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.
Harvey Dubner, Repunit R49081 is a probable prime, Math. Comp., 71 (2001), 833-835.
Graham, Knuth and Patashnik, Concrete Mathematics, Addison-Wesley, 1994; see p 146 problem 22. [From Cino Hilliard (hillcino368(AT)hotmail.com), Dec 23 2008]
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).
H. C. Williams and Harvey Dubner, The primality of R1031, Math. Comp., 47(176), Oct 1986, 703-711.
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LINKS
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Table of n, a(n) for n=1..9.
Harvey Dubner, New probable prime repunit, R(49081), Number Theory List, Sep 09 1999.
Martianus Frederic Ezerman, Bertrand Meyer, and Patrick Sole, On Polynomial Pairs of Integers, 2012. - From N. J. A. Sloane, Nov 08 2012
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, Posting to Number Theory List : Apr 03 2007
Makoto Kamada, Factorizations of 11...11 (Repunit).
H. Lifchitz, Mersenne and Fermat primes field
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; 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
Index entries for primes involving repunits
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EXAMPLE
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2 appears because the 2-digit repunit 11 = eleven is prime. 19 appears because the 19-digit repunit 1111111111111111111 is prime.
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MATHEMATICA
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lst={}; Do[p=(10^n-1)/9; If[PrimeQ[p], Print[q=Length[IntegerDigits[p]]]; AppendTo[lst, q]], {n, 0, 8!}]; lst [From Vladimir Joseph Stephan Orlovsky, Sep 28 2008]
Select[Range[271000], PrimeQ[FromDigits[PadRight[{}, #, 1]]]&] (* From Harvey P. Dale, Nov 05 2011 *)
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PROG
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(PARI) forprime(x=2, 20000, if(ispseudoprime((10^x-1)/9), print1(x", "))) \\ Cino Hilliard (hillcino368(AT)hotmail.com), Dec 23 2008
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CROSSREFS
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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
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KEYWORD
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hard,nonn,nice,more
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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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 (paul.bourdelais(AT)gd-ais.com) and Harvey Dubner (harvey(AT)dubner.com) around Mar 26-28 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
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STATUS
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approved
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