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A004022
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Primes of the form (10^k - 1)/9. Also called repunit primes or repdigit primes.
(Formerly M4816)
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111
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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 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
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
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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.
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).
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LINKS
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J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
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FORMULA
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MATHEMATICA
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Select[Table[FromDigits[PadRight[{}, n, 1]], {n, 30}], PrimeQ] (* Harvey P. Dale, Apr 07 2018 *)
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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, 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))
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CROSSREFS
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KEYWORD
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nonn,nice,bref
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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