

A005385


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


246



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, 2963
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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
Primes p such that p1 is a semiprime.  Zak Seidov, Jul 01 2005
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
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
From the fourth entry onward, do these correspond to Smarandache's problem 34 (see A007931 link), specifically values which cannot be used (do not meet conditions) to confirm the conjecture?  Bill McEachen, Sep 29 2016
Primes p with the property that there is a prime q such that p+q^2 is a square.  Zak Seidov, Feb 16 2017
It is conjectured that there are infinitely many safe primes, and their estimated asymptotic density ~ 2C/(log n)^2 (where C = 0.66... is the twin prime constant A005597) converges to the actual value as far as we know.  M. F. Hasler, Jun 14 2021


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

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].


FORMULA



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
(PARI) [xx<primes(10^3), bigomega(x1)==2] \\ Altug Alkan, Nov 04 2015
(Haskell)
a005385 n = a005385_list !! (n1)
a005385_list = filter ((== 1) . a010051 . (`div` 2)) a000040_list
(Magma) [p: p in PrimesUpTo(3000)  IsPrime((p1) div 2)]; // Vincenzo Librandi, Jul 06 2015
(Python)
from sympy import isprime, primerange
def aupto(limit):
alst = []
for p in primerange(1, limit+1):
if isprime((p1)//2): alst.append(p)
return alst


CROSSREFS

Except for the initial term, this is identical to A079148.


KEYWORD

nonn,easy,nice


AUTHOR



EXTENSIONS

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


STATUS

approved



