1,1

The indices of primes with digital product (i.e. product of digits) equal to 1.

The larger terms may only correspond to probable primes.

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

These indices p must also be prime. If p is not prime, say p=mn, then 10^mn-1

=((10^m)^n)-1 => 10^m-1 divides 10^mn-1. Since 9 divides 10^m-1 or (10^m-1)/9

= q, it follows q divides (10^p-1)/9. This is a result of the identity,

a^n-b^n = (a-b)(a^(n-1) + a^(n-2)b + . . . + b^(n-1). (End)

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

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 19, pp 6, Ellipses, Paris 2008.

Harvey Dubner, New probable prime repunit R(49081), posting to Number Theory List (NMBRTHRY(AT)LISTSERV.NODAK.EDU) Sep 09, 1999.

Harvey Dubner, Repunit R49081 is a probable prime, Math. Comp., 71 (2001), 833-835.

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

H. C. Williams and Harvey Dubner, The primality of R1031, Math. Comp., 47(176), Oct 1986, 703-711.

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

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

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

K. S. Brown's Mathpages, Seeking Prime Repunits

C. K. Caldwell, The Prime Glossary, Repunit

Patrick De Geest, Circular Primes

H. Dubner, Posting to Number Theory List : 3 April 2007

Makoto Kamada, Factorizations of 11...11 (Repunit).

H. Lifchitz, Mersenne and Fermat primes field

Andy Steward, Prime Generalized Repunits

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

E. Wegrzynowski, Nombres 1_[n] premiers

Eric Weisstein's World of Mathematics, Repunit

Eric Weisstein's World of Mathematics, Integer Sequence Primes

Index entries for primes involving repunits

2 appears because the 2-digit repunit 11 = eleven is prime. 19 appears because the 19-digit repunit 1111111111111111111 is prime.

lst={}; Do[p=(10^n-1)/9; If[PrimeQ[p], Print[q=Length[IntegerDigits[p]]]; AppendTo[lst, q]], {n, 0, 8!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Sep 28 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(x", ")))

(End)

See A004022 for the actual primes.

Cf. A055557, A002275.

Sequence in context: A037003 A105907 A018696 this_sequence A031030 A083689 A102617

Adjacent sequences: A004020 A004021 A004022 this_sequence A004024 A004025 A004026

hard,nonn,nice

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

49081 found by Harvey Dubner - posting to Number Theory List (NMBRTHRY(AT)LISTSERV.NODAK.EDU) Sep 09, 1999.

86453 found using pfgw (a faster version of PrimeForm) on Oct 26 2000 by Lew Baxter (ldenverb(AT)hotmail.com) - posting to Number Theory List (NMBRTHRY(AT)LISTSERV.NODAK.EDU), Oct 26, 2000.

a(8) = 109297 was apparently discovered independently by (in alphabetical order) Paul Bourdelais (paul.bourdelais(AT)gd-ais.com) and Harvey Dubner (harvey(AT)dubner.com) around Mar 26-28 2007.

A new probable prime repunit, R(270343), was found Jul 11 2007 by Maksym Voznyy (mvoznyy0526(AT)ROGERS.COM) and Anton Budnyy.