

A005385


Safe primes p: (p1)/2 is also prime.
(Formerly M3761)


184



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

1,1


COMMENTS

Then (p1)/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(p1) + 1/p)=6.  Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Feb 10 2004
Or, primes p such that p1 is a semiprime.  Zak Seidov, Jul 01 2005
A156659(a(n)) = 1; A156875 gives numbers of safe primes <= n.  Reinhard Zumkeller, Feb 18 2009
Near subset of A161897.  Reikku Kulon, Jun 21 2009
From Daniel Forgues, Jul 31 2009: (Start)
A safe prime p is 7 or of the form 6k1, 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 (p1)/2 is composite and divisible by 3. (End)
If k is the product of the nth safe prime p and its corresponding Sophie Germain prime (p1)/2, then a(n) = 2(kphi(k))/3 + 1, where phi is Euler's totient function.  Wesley Ivan Hurt, Oct 03 2013
From Bob Selcoe, Apr 14 2014: (Start)
When the nth prime is divided by all primes up to the (n1)th prime, safe primes (p) have remainders of 1 when divided by 2 and (p1)/2 and no other primes. That is, p(mod j)=1 iff j={2,(p1)/2}; p>j, {p,j}=>prime. Explanation: Generally, x(mod y)=1 iff x=y'+1, where y' is the set of divisors of y, y'>1. Since safe primes (p) are of the form p(mod j)=1 iff p and j are prime, then j={j'}. That is, since j is prime, there are no divisors of j (greater than 1) other than j. Therefore, no primes other than j exist which satisfy the equation p(mod j)=1.
Except primes of the form 2^n+1 (n>=0), all nonsafe primes (p') will have at least one prime (p") greater than 2 and less than (p1)/2 such that p'(mod p")=1. Explanation: Nonsafe primes (p') are of the form p'(mod k)=1 where k is composite. This means prime divisors of k exist, and p" is the set of prime divisors of k (example p'=89: k=44; p"={2,11}). The exception applies because p"={2} iff p'=2^n+1.
Refer to the rows in triangle A207409 for illustration and further explanation. (End)
Conjecture: there is a strengthening of the Bertrand postulate for n>=24: the interval (n, 2*n) contains a safe prime. It has been tested by Peter J. C. Moses up to n = 10^7.  Vladimir Shevelev, Jul 06 2015
The six known safe primes p such that (p1)/2 is a Fibonacci prime are in A263880.  Jonathan Sondow, Nov 04 2015


REFERENCES

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.
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..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].
B. Cloitre, On the fractal behavior of primes, 2011.
David Naccache, DoubleSpeed Safe Prime Generation
Planetmath, Safe prime
Michael J. Wiener, Safe Prime Generation with a Combined Sieve
Wikipedia, Safe prime


FORMULA

a(n) = 2 * A005384(n) + 1.
a(n) = A077065(n) + 1.  Paolo P. Lava, Jun 10 2013


MAPLE

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])): # Peter Luschny, Nov 08 2010


MATHEMATICA

Select[Prime[Range[1000]], PrimeQ[(#1)/2]&] (* Zak Seidov, Jan 26 2011 *)


PROG

(PARI) g(n) = forprime(x=2, n, y=x+x+1; if(isprime(y), print1(y", "))) \\ Cino Hilliard, Sep 12 2004
(Haskell)
a005385 n = a005385_list !! (n1)
a005385_list = filter ((== 1) . a010051 . (`div` 2)) a000040_list
 Reinhard Zumkeller, Sep 18 2011
(MAGMA) [p: p in PrimesUpTo(3000)  IsPrime((p1) div 2)]; // Vincenzo Librandi, Jul 06 2015
(PARI) [xx<primes(10^3), bigomega(x1)==2] \\ Altug Alkan, Nov 04 2015


CROSSREFS

Cf. A007700, A023272, A023302, A023330, A057331, A005602, A207409, A263880.
Except for the initial term, this is identical to A079148.
Cf. A161897.
Subsequence of A088707.
Primes in A072055.
Sequence in context: A226027 A090810 A092307 * A181602 A075705 A141305
Adjacent sequences: A005382 A005383 A005384 * A005386 A005387 A005388


KEYWORD

nonn,easy,nice


AUTHOR

N. J. A. Sloane


EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), Feb 15 2001


STATUS

approved



