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A055557
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Numbers k such that 3*R_k - 2 is prime, where R_k = 11...1 is the repunit (A002275) of length k.
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8
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2, 3, 4, 5, 6, 7, 8, 18, 40, 50, 60, 78, 101, 151, 319, 382, 784, 1732, 1918, 8855, 11245, 11960, 12130, 18533, 22718, 23365, 24253, 24549, 25324, 30178, 53718, 380976, 424861
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OFFSET
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1,1
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COMMENTS
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Also numbers k such that (10^k-7)/3 is prime.
Sierpiński attributes the primes for k = 2,...,8 to A. Makowski.
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.)
Concerning certifying primes, see the references by Goldwasser et al., Atkin et al. and Morain. - Labos
No more than 14 consecutive exponents can provide primes because for exponents 15m+2, 16m+9, 18m+12, 22m+21, terms are divisible by 31, 17, 19, 23 respectively. Here 7 of possible 14 is realized. - Labos Elemer, Jan 19 2005
(10^(15m+2)-7)/3 == 0 (mod 31). So 15m+2 isn't a term for m > 0. - Seiichi Manyama, Nov 05 2016
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REFERENCES
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C. Caldwell, The near repdigit primes 333...331, J.Recreational Math. 21:4 (1989) 299-304.
S. Goldwasser and J. Kilian, Almost All Primes Can Be Quickly Certified. in Proc. 18th STOC, 1986, pp. 316-329.
W. Sierpiński, 200 Zadan z Elementarnej Teorii Liczb [200 Problems from the Elementary Theory of Numbers], Warszawa, 1964; Problem 88.
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LINKS
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Table of n, a(n) for n=1..33.
A. Atkin and F. Morain, Elliptic Curves and Primality Proving, Math. Comp. 61:29-68, 1993.
Makoto Kamada, Prime numbers of the form 33...331.
Mathematics.StackExchange.com, 31,331,3331, 33331,333331,3333331,33333331 are prime
F. Morain, Implementation of the Atkin-Goldwasser-Kilian Primality Testing Algorithm, INRIA Research Report, # 911, October 1988.
Dave Rusin, Primes in exponential sequences [Broken link]
Dave Rusin, Primes in exponential sequences [Cached copy]
Index entries for primes involving repunits
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FORMULA
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a(n) = A055520(n) + 1.
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MATHEMATICA
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Do[ If[ PrimeQ[(10^n - 7)/3], Print[n]], {n, 50410}]
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 Elemer *)
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PROG
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(PARI) for(n=1, 2000, if(isprime((10^n-7)/3), print(n)))
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CROSSREFS
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Cf. A051200, A033175, A055520.
Sequence in context: A250043 A260354 A114799 * A173577 A103205 A039127
Adjacent sequences: A055554 A055555 A055556 * A055558 A055559 A055560
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KEYWORD
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nonn
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AUTHOR
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Labos Elemer, Jul 10 2000
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EXTENSIONS
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Corrected and extended by Jason Earls, Sep 22 2001
a(20)-a(31) were found by Makoto Kamada (see links for details). At present they correspond only to probable primes.
a(32)-a(33) from Leonid Durman, Jan 09-10 2012
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STATUS
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approved
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