|
| |
|
|
A055557
|
|
Numbers n such that 3*R_n - 2 is prime, where R_n = 11...1 is the repunit (A002275) of length n.
|
|
6
| |
|
|
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
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,1
|
|
|
COMMENTS
| Also numbers n such that (10^n-7)/3 is prime.
Sierpinski attributes the primes for n = 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 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
|
|
|
REFERENCES
| A. O. L. Atkin and F Morain: Elliptic Curves and Primality Proving, Mathematics of Computation, 1993, 29-68.
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.
F. Morain in "Implementation of the Atkin-Goldwasser-Kilian Primality Testing Algorithm", INRIA Research Report, # 911, October 1988.
W. Sierpinski, 200 Zadan z Elementarnej Teorii Liczb [200 Problems from the Elementary Theory of Numbers], Warszawa, 1964; Problem 88.
|
|
|
LINKS
| Makoto Kamada, Factorizations of near-repdigit numbers
Makoto Kamada, News and updates, October 2004
Makoto Kamada, Factorizations of 33...331
Dave Rusin, Primes in exponential sequences
Index entries for primes involving repunits
|
|
|
MATHEMATICA
| 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)
|
|
|
PROG
| (PARI) for(n=1, 2000, if(isprime((10^n-7)/3), print(n)))
|
|
|
CROSSREFS
| Cf. A051200, A033175.
Equals A055520 plus 1.
Sequence in context: A171723 A039050 A114799 * A173577 A103205 A039127
Adjacent sequences: A055554 A055555 A055556 * A055558 A055559 A055560
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Labos E. (labos(AT)ana.sote.hu), Jul 10 2000
|
|
|
EXTENSIONS
| Corrected and extended by Jason Earls (zevi_35711(AT)yahoo.com), Sep 22 2001
a[20]-a[31] were found by Makoto Kamada (see links for details). At present a[20]-a[31] correspond only to probable primes.
a(32)-a(33) from Leonid Durman, Jan 09-10 2012
|
| |
|
|
