1,1

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

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.

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

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)

(PARI) for(n=1, 2000, if(isprime((10^n-7)/3), print(n)))

Cf. A051200, A033175.

Equals A055520 plus 1.

Sequence in context: A066759 A039050 A114799 this_sequence A103205 A039127 A024650

Adjacent sequences: A055554 A055555 A055556 this_sequence A055558 A055559 A055560

nonn

Labos E. (labos(AT)ana.sote.hu), Jul 10 2000

Corrected and extended by Jason Earls (jcearls(AT)cableone.net), 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.