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OFFSET
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1,1
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COMMENTS
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The next term corresponds to n = 317 and is too large to include: see A004023, A046413.
Also called repunit primes or prime repunits.
Also, primes with digital product = 1.
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)
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REFERENCES
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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]
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).
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.
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LINKS
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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
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FORMULA
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a(n) = A002275(A004023(n)).
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MATHEMATICA
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lst={}; Do[If[PrimeQ[p = (10^n - 1)/9], AppendTo[lst, p]], {n, 10^2}]; lst (* From Vladimir Orlovsky, Aug 22 2008 *)
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PROG
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(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
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CROSSREFS
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See A004023 for the number of 1's. Cf. A046413.
Sequence in context: A198244 A066953 A213645 * A083344 A063863 A187795
Adjacent sequences: A004019 A004020 A004021 * A004023 A004024 A004025
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KEYWORD
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nonn,nice,bref
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Edited by Max Alekseyev, Nov 15 2010
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STATUS
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approved
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