1,1

Also called repunit primes.

The prime repunits, primes with digital product = 1.

The next term is too large to include: see A004023, A046413.

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

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)

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

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 11.

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.

Graham, Knuth and Patashnik, Concrete mathematics, Addison-Wesley, 1994; see p 146 problem 22. [From Cino Hilliard (hillcino368(AT)hotmail.com), Dec 23 2008]

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

T. D. Noe, Table of n, a(n) for n=1..5

J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.

Andy Steward, Prime Generalized Repunits

S. S. Wagstaff, Jr., The Cunningham Project

lst={}; Do[If[PrimeQ[p=(10^n-1)/9], AppendTo[lst, p]], {n, 10^2}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Aug 22 2008]

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

(PARI) forprime(x=2, 20000, if(ispseudoprime((10^x-1)/9), print1((10^x-1)/9", ")))

(End)

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

Adjacent sequences: A004019 A004020 A004021 this_sequence A004023 A004024 A004025

Sequence in context: A072218 A046844 A066953 this_sequence A083344 A063863 A101364

nonn,nice,bref

N. J. A. Sloane (njas(AT)research.att.com).