The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A004022 Primes of the form (10^k - 1)/9. Also called repunit primes or repdigit primes. (Formerly M4816) 114
 11, 1111111111111111111, 11111111111111111111111 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS The next term corresponds to k = 317 and is too large to include: see A004023. Also called repunit primes or prime repunits. Also, primes with digital product = 1. The number of 1's in these repunits must also be prime. Since the number of 1's in (10^k-1)/9 is k, if k = p*m then (10^(p*m)-1) = (10^p)^m-1 => (10^p-1)/9 = q and q divides (10^k-1). This follows from the identity a^k - b^k = (a-b)*(a^(k-1) + a^(k-2)*b + ... + b^(k-1)). - Cino Hilliard, Dec 23 2008 A subset of A020449, ..., A020457, A036953, ..., cf. link to OEIS index. - M. F. Hasler, Jul 27 2015 The terms in this sequence, except 11 which is not Brazilian, are prime repunits in base ten, so they are Brazilian primes belonging to A085104 and A285017. - Bernard Schott, Apr 08 2017 REFERENCES 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. 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. Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 60. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS 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. Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33. Makoto Kamada, Factorizations of 11...11 (Repunit). D. H. Lehmer, On the number (10^23-1)/9, Bull. Amer. Math. Soc. 35 (1929), 349-350. James Maynard and Brady Haran, Primes without a 7, Numberphile video (2019) Andy Steward, Prime Generalized Repunits S. S. Wagstaff, Jr., The Cunningham Project Index to entries for primes with digits in a given set. FORMULA a(n) = A002275(A004023(n)). MATHEMATICA lst={}; Do[If[PrimeQ[p = (10^n - 1)/9], AppendTo[lst, p]], {n, 10^2}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 22 2008 *) Select[Table[(10^n - 1) / 9, {n, 500}], PrimeQ] (* Vincenzo Librandi, Nov 08 2014 *) Select[Table[FromDigits[PadRight[{}, n, 1]], {n, 30}], PrimeQ] (* Harvey P. Dale, Apr 07 2018 *) PROG (PARI) forprime(x=2, 20000, if(ispseudoprime((10^x-1)/9), print1((10^x-1)/9", "))) \\ Cino Hilliard, Dec 23 2008 (Magma) [a: n in [0..300] | IsPrime(a) where a is (10^n - 1) div 9 ]; // Vincenzo Librandi, Nov 08 2014 (Python) from sympy import isprime from itertools import count, islice def agen(): # generator of terms yield from (t for t in (int("1"*k) for k in count(1)) if isprime(t)) print(list(islice(agen(), 4))) # Michael S. Branicky, Jun 09 2022 CROSSREFS A116692 is another version of repunit primes or repdigit primes. - N. J. A. Sloane, Jan 22 2023 See A004023 for the number of 1's. Cf. A046413. Sequence in context: A257139 A257166 A257167 * A243534 A257304 A241570 Adjacent sequences: A004019 A004020 A004021 * A004023 A004024 A004025 KEYWORD nonn,nice,bref AUTHOR N. J. A. Sloane EXTENSIONS Edited by Max Alekseyev, Nov 15 2010 Name expanded by N. J. A. Sloane, Jan 22 2023 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified August 13 22:54 EDT 2024. Contains 375146 sequences. (Running on oeis4.)