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A005385
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Safe primes p: (p-1)/2 is also prime.
(Formerly M3761)
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171
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5, 7, 11, 23, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619, 1823, 1907, 2027, 2039, 2063, 2099, 2207, 2447, 2459, 2579, 2819, 2879, 2903
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OFFSET
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1,1
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COMMENTS
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Then (p-1)/2 is called a Sophie Germain prime: see A005384.
Or, primes of the form 2p+1 where p is prime.
Primes p such that denominator(Bernoulli(p-1) + 1/p)=6. - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Feb 10 2004
Or, primes p such that p-1 is a semiprime. [From Zak Seidov, Jul 01 2005]
A156659(a(n)) = 1; A156875 gives numbers of safe primes <= n. [From Reinhard Zumkeller, Feb 18 2009]
Near subset of A161897. [From Reikku Kulon, Jun 21 2009]
Contribution from Daniel Forgues, Jul 31 2009: (Start)
A safe prime p is 7 or of the form 6k-1, k >= 1, i.e. p == 5 (mod 6).
A prime p of the form 6k+1, k >= 2, i.e. p = 1 (mod 6), cannot be a safe prime since (p-1)/2 is composite and divisible by 3. (End)
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
B. Cloitre, On the fractal behavior of primes, 2011; http://bcmathematics.monsite-orange.fr/FractalOrderOfPrimes.pdf
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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T. D. Noe, Table of n, a(n) for n=1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
David Naccache, Double-Speed Safe Prime Generation
Planetmath, Safe prime
Michael J. Wiener, Safe Prime Generation with a Combined Sieve
Wikipedia, Safe prime
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FORMULA
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a(n) = 2 * A005384(n) + 1.
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MAPLE
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with(numtheory); [ seq(safeprime(i), i=1..3000) ]: convert(%, set); convert(%, list); sort(%);
A005385_list := n->select(i->isprime(iquo(i, 2)), select(i->isprime(i), [$1..n])): [From Peter Luschny, Nov 08 2010]
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MATHEMATICA
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Select[Prime[Range[1000]], PrimeQ[(#-1)/2]&] (* From Zak Seidov, Jan 26 2011 *)
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PROG
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(PARI) g(n) = forprime(x=2, n, y=x+x+1; if(isprime(y), print1(y", "))) (Cino Hilliard (hillcino368(AT)gmail.com), Sep 12 2004)
(Haskell)
a005385 n = a005385_list !! (n-1)
a005385_list = filter ((== 1) . a010051 . (`div` 2)) a000040_list
-- Reinhard Zumkeller, Sep 18 2011
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CROSSREFS
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Cf. A007700, A023272, A023302, A023330, A057331, A005602.
Except for the initial term, this is identical to A079148.
Cf. A161897 [From Reikku Kulon, Jun 21 2009]
Subsequence of A088707.
Subsequence of A072055. - Zak Seidov, Mar 12 2013
Sequence in context: A226027 A090810 A092307 * A181602 A075705 A141305
Adjacent sequences: A005382 A005383 A005384 * A005386 A005387 A005388
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Feb 15 2001
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STATUS
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approved
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