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 A004022 Primes of form (10^n - 1)/9. (Formerly M4816) 68

%I M4816

%S 11,1111111111111111111,11111111111111111111111

%N Primes of form (10^n - 1)/9.

%C The next term corresponds to n = 317 and is too large to include: see A004023, A046413.

%C Also called repunit primes or prime repunits.

%C Also, primes with digital product = 1.

%C Contribution from Cino Hilliard (hillcino368(AT)hotmail.com), Dec 23 2008: (Start)

%C The number of 1's in these repunits must also be prime. Since the number of 1's in (10^n-1)/9 is n, if n = pk then (10^pk-1)=(10^p)^k-1 => (10^p-1)/9 = q and q divides (10^n-1). This follows from the identity a^n-b^n=(a-b)(a^(n-1)+a^(n-2)b+...+b^n-1). (End)

%D T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, p. 11. Graham, Knuth and Patashnik, Concrete mathematics, Addison-Wesley, 1994; see p. 146, problem 22. [From Cino Hilliard (hillcino368(AT)hotmail.com), Dec 23 2008]

%D Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 60.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D M. Barsanti, R. Dvornicich, M. Forti, T. Franzoni, M. Gobbino, S. Mortola, L. Pernazza and R. Romito, Il Fibonacci N. 8 (included in Il Fibonacci, Unione Matematica Italiana, 2011), 2004, Problem 8.10.

%H T. D. Noe, <a href="/A004022/b004022.txt">Table of n, a(n) for n = 1..5</a>

%H 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 Andy Steward, <a href="http://www.users.globalnet.co.uk/~aads/primes.html">Prime Generalized Repunits</a>

%H S. S. Wagstaff, Jr., <a href="http://www.cerias.purdue.edu/homes/ssw/cun/index.html">The Cunningham Project</a>

%F a(n) = A002275(A004023(n)).

%t lst={}; Do[If[PrimeQ[p = (10^n - 1)/9], AppendTo[lst, p]], {n, 10^2}]; lst (* From Vladimir Orlovsky, Aug 22 2008 *)

%o (PARI) forprime(x=2,20000,if(ispseudoprime((10^x-1)/9),print1((10^x-1)/9","))) \\ Cino Hilliard (hillcino368(AT)hotmail.com), Dec 23 2008

%Y See A004023 for the number of 1's. Cf. A046413.

%K nonn,nice,bref

%O 1,1

%A _N. J. A. Sloane_.

%E Edited by Max Alekseyev, Nov 15 2010

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