%I A004023 M2114
%S A004023 2,19,23,317,1031,49081,86453,109297
%N A004023 Indices of prime "repunits": numbers n such that 11...111 = (10^n - 1)/
               9 is prime.
%C A004023 The indices of primes with digital product (i.e. product of digits) equal 
               to 1.
%C A004023 The larger terms may only correspond to probable primes.
%C A004023 Contribution from Cino Hilliard (hillcino368(AT)hotmail.com), Dec 23 
               2008: (Start)
%C A004023 These indices p must also be prime. If p is not prime, say p=mn, then 
               10^mn-1
%C A004023 =((10^m)^n)-1 => 10^m-1 divides 10^mn-1. Since 9 divides 10^m-1 or (10^m-1)/
               9
%C A004023 = q, it follows q divides (10^p-1)/9. This is a result of the identity,
%C A004023 a^n-b^n = (a-b)(a^(n-1) + a^(n-2)b + . . . + b^(n-1). (End)
%D A004023 J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, 
               Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and 
               later supplements.
%D A004023 J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 19, pp 6, Ellipses, 
               Paris 2008.
%D A004023 Harvey Dubner, New probable prime repunit R(49081), posting to Number 
               Theory List (NMBRTHRY(AT)LISTSERV.NODAK.EDU) Sep 09, 1999.
%D A004023 Harvey Dubner, Repunit R49081 is a probable prime, Math. Comp., 71 (2001), 
               833-835.
%D A004023 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A004023 H. C. Williams and Harvey Dubner, The primality of R1031, Math. Comp., 
               47(176), Oct 1986, 703-711.
%D A004023 Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 
               60.
%D A004023 Grahm,Knuth,Patashnik, Concrete Mathematics, Addison-Wesley, 1994; see 
               p 146 problem 22. [From Cino Hilliard (hillcino368(AT)hotmail.com), 
               Dec 23 2008]
%H A004023 J. Brillhart et al., <a href="http://www.ams.org/online_bks/conm22/">
               Factorizations of b^n +- 1</a>, Contemporary Mathematics, Vol. 22, 
               Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
%H A004023 K. S. Brown's Mathpages, <a href="http://www.mathpages.com/home/kmath317.htm">
               Seeking Prime Repunits</a>
%H A004023 C. K. Caldwell, The Prime Glossary, <a href="http://primes.utm.edu/glossary/
               page.php?sort=Repunit">Repunit</a>
%H A004023 Patrick De Geest, <a href="http://www.worldofnumbers.com/circular.htm">
               Circular Primes</a>
%H A004023 H. Dubner, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0704&L=nmbrthry&T=0&P=178">
               Posting to Number Theory List : 3 April 2007</a>
%H A004023 Makoto Kamada, <a href="http://homepage2.nifty.com/m_kamada/math/11111.htm">
               Factorizations of 11...11 (Repunit)</a>.
%H A004023 H. Lifchitz, <a href="http://www.primenumbers.net/Henri/us/MersFermus.htm">
               Mersenne and Fermat primes field</a>
%H A004023 Andy Steward, <a href="http://www.users.globalnet.co.uk/~aads/primes.html">
               Prime Generalized Repunits</a>
%H A004023 S. S. Wagstaff, Jr., <a href="http://www.cerias.purdue.edu/homes/ssw/
               cun/index.html">The Cunningham Project</a>
%H A004023 E. Wegrzynowski, <a href="http://www.lifl.fr/~wegrzyno/RepunitBase10/
               repbase101.html">Nombres 1_[n] premiers</a>
%H A004023 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Repunit.html">Repunit</a>
%H A004023 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               IntegerSequencePrimes.html">Integer Sequence Primes</a>
%H A004023 <a href="Sindx_Pri.html#Pri_rep">Index entries for primes involving repunits</
               a>
%e A004023 2 appears because the 2-digit repunit 11 = eleven is prime. 19 appears 
               because the 19-digit repunit 1111111111111111111 is prime.
%t A004023 lst={};Do[p=(10^n-1)/9;If[PrimeQ[p], Print[q=Length[IntegerDigits[p]]];
               AppendTo[lst, q]], {n, 0, 8!}];lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), 
               Sep 28 2008]
%o A004023 Contribution from Cino Hilliard (hillcino368(AT)hotmail.com), Dec 23 
               2008: (Start)
%o A004023 (PARI) forprime(x=2,20000,if(ispseudoprime((10^x-1)/9),print1(x",")))
%o A004023 (End)
%Y A004023 See A004022 for the actual primes.
%Y A004023 Cf. A055557, A002275.
%Y A004023 Sequence in context: A037003 A105907 A018696 this_sequence A031030 A083689 
               A102617
%Y A004023 Adjacent sequences: A004020 A004021 A004022 this_sequence A004024 A004025 
               A004026
%K A004023 hard,nonn,nice
%O A004023 1,1
%A A004023 N. J. A. Sloane (njas(AT)research.att.com).
%E A004023 49081 found by Harvey Dubner - posting to Number Theory List (NMBRTHRY(AT)LISTSERV.NODAK.EDU) 
               Sep 09, 1999.
%E A004023 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.
%E A004023 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.
%E A004023 A new probable prime repunit, R(270343), was found Jul 11 2007 by Maksym 
               Voznyy (mvoznyy0526(AT)ROGERS.COM) and Anton Budnyy.