Search: id:A055557
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%I A055557
%S A055557 2,3,4,5,6,7,8,18,40,50,60,78,101,151,319,382,784,1732,1918,8855,11245,
%T A055557 11960,12130,18533,22718,23365,24253,24549,25324,30178,53718
%N A055557 Numbers n such that 3*R_n - 2 is prime, where R_n = 11...1 is the repunit 
               (A002275) of length n.
%C A055557 Also numbers n such that (10^n-7)/3 is prime.
%C A055557 Sierpinski attributes the primes for n = 2,...,8 to A. Makowski.
%C A055557 The history of the discovery of these numbers may be as follows: a(1)-a(7), 
               Makowski; a(8)-a(18), Caldwell; a(19), Earls; a(20)-a(31), Kamada. 
               (Corrections to this account will be welcomed.)
%C A055557 Concerning certifying primes, see the references by Goldwasser et al., 
               Atkin et al. and Morain. - Labos
%C A055557 No more than 14 consecutive exponents can provide primes because for 
               exponents 15k+2, 16k+9, 18k+12, 22k+21, a(n)'s are divisible by 31, 
               17, 19, 23 respectively. Here 7 of possible 14 is realized. - Labos 
               E. (labos(AT)ana.sote.hu), Jan 19 2005
%D A055557 A. O. L. Atkin and F Morain: Elliptic Curves and Primality Proving, Mathematics 
               of Computation, 1993, 29-68.
%D A055557 C. Caldwell: The near repdigit primes 333...331, J.Recreational Math. 
               21:4 (1989) 299-304.
%D A055557 S. Goldwasser and J. Kilian: Almost All Primes Can Be Quickly Certified. 
               in Proc. 18th STOC, 1986, pp. 316-329.
%D A055557 F. Morain in "Implementation of the Atkin-Goldwasser-Kilian Primality 
               Testing Algorithm", INRIA Research Report, # 911, October 1988.
%D A055557 W. Sierpinski, 200 Zadan z Elementarnej Teorii Liczb [200 Problems from 
               the Elementary Theory of Numbers], Warszawa, 1964; Problem 88.
%H A055557 Makoto Kamada, <a href="http://homepage2.nifty.com/m_kamada/math/factorizations.htm">
               Factorizations of near-repdigit numbers</a>
%H A055557 Makoto Kamada, <a href="http://homepage2.nifty.com/m_kamada/math/nu200410.htm">
               News and updates, October 2004</a>
%H A055557 Makoto Kamada, <a href="http://homepage2.nifty.com/m_kamada/math/33331.htm">
               Factorizations of 33...331</a>
%H A055557 Dave Rusin, <a href="http://www.math.niu.edu/~rusin/known-math/98/exp_primes">
               Primes in exponential sequences</a>
%H A055557 <a href="Sindx_Pri.html#Pri_rep">Index entries for primes involving repunits</
               a>
%t A055557 Do[ If[ PrimeQ[(10^n - 7)/3], Print[n]], {n, 50410}]
%t A055557 One may run the prime certificate program as follows <<NumberTheory`PrimeQ` 
               Table[{n, ProvablePrimeQ[(-7+10^Part[t, n])/3, Certificate->True]}, 
               {n, 1, 16}] (Labos)
%o A055557 (PARI) for(n=1,2000, if(isprime((10^n-7)/3),print(n)))
%Y A055557 Cf. A051200, A033175.
%Y A055557 Equals A055520 plus 1.
%Y A055557 Sequence in context: A066759 A039050 A114799 this_sequence A103205 A039127 
               A024650
%Y A055557 Adjacent sequences: A055554 A055555 A055556 this_sequence A055558 A055559 
               A055560
%K A055557 nonn
%O A055557 1,1
%A A055557 Labos E. (labos(AT)ana.sote.hu), Jul 10 2000
%E A055557 Corrected and extended by Jason Earls (zevi_35711(AT)yahoo.com), Sep 
               22 2001
%E A055557 a[20]-a[31] were found by Makoto Kamada (see links for details). At present 
               a[20]-a[31] correspond only to probable primes.

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