A156875 -id:A156875 - cp's OEIS frontend

A005385 Safe primes p: (p-1)/2 is also prime.

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

Offset: 1

N. J. A. Sloane

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
Primes p such that p-1 is a semiprime. - Zak Seidov, Jul 01 2005
A156659(a(n)) = 1; A156875 gives numbers of safe primes <= n. - Reinhard Zumkeller, Feb 18 2009
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)
If k is the product of the n-th safe prime p and its corresponding Sophie Germain prime (p-1)/2, then a(n) = 2(k-phi(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 n-th prime is divided by all primes up to the (n-1)-th prime, safe primes (p) have remainders of 1 when divided by 2 and (p-1)/2 and no other primes. That is, p(mod j)=1 iff j={2,(p-1)/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 non-safe primes (p') will have at least one prime (p") greater than 2 and less than (p-1)/2 such that p'(mod p")=1. Explanation: Non-safe 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 (p-1)/2 is a Fibonacci prime are in A263880. - Jonathan Sondow, Nov 04 2015
The only term in common with A005383 is 5. - Zak Seidov, Dec 31 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

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).

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].
A. R. Ashrafi and F. Koorepazan-Moftakhar, Towards the Classification of Finite Simple Groups with exactly Three or Four Supercharacter Theories, arXiv preprint arXiv:1605.08971 [math.GR], 2016.
R. P. Boas & N. J. A. Sloane, Correspondence, 1974.
Siji Chen and Sheng Chen, Connectedness of digraphs from quadratic polynomials, Involve (2020) Vol. 13, No. 2, 357-360.
Benoit Cloitre, On the fractal behavior of primes, 2011.
L. H. Gallardo and O. Rahavandrainy, There are finitely many even perfect polynomials over F_p with p+1 irreducible divisors, Acta Mathematica Universitatis Comenianae, Vol. 83, No. 2, 2016, 261-275.
David Naccache, Double-Speed Safe Prime Generation, IACR, Report 2003/175, 2003.
Planetmath, Safe prime.
Michael J. Wiener, Safe Prime Generation with a Combined Sieve, IACR, Report 2003/186, 2003.
Wikipedia, Safe prime.

Cf. A007700, A023272, A023302, A023330, A057331, A005602, A207409, A263880.
Except for the initial term, this is identical to A079148.
Cf. A005384, A005383.
Subsequence of A088707.
Primes in A072055.

a005385 n = a005385_list !! (n-1)
a005385_list = filter ((== 1) . a010051 . (`div` 2)) a000040_list
-- Reinhard Zumkeller, Sep 18 2011

[p: p in PrimesUpTo(3000) | IsPrime((p-1) div 2)]; // Vincenzo Librandi, Jul 06 2015

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

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

g(n) = forprime(x=2,n,y=x+x+1;if(isprime(y),print1(y","))) \\ Cino Hilliard, Sep 12 2004

[x|x<-primes(10^3), bigomega(x-1)==2] \\ Altug Alkan, Nov 04 2015

from sympy import isprime, primerange
def aupto(limit):
  alst = []
  for p in primerange(1, limit+1):
    if isprime((p-1)//2): alst.append(p)
  return alst
print(aupto(2963)) # Michael S. Branicky, May 07 2021

a(n) = 2 * A005384(n) + 1.

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

A156659 Characteristic function of safe primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Reinhard Zumkeller, Feb 13 2009

nonn

R. Zumkeller, Table of n, a(n) for n = 0..10000
Wikipedia, Safe prime
Index entries for characteristic functions

Cf. A005385, A156660, A292936.

a156659 n = fromEnum $ a010051 n == 1 && a010051 (n `div` 2) == 1
-- Reinhard Zumkeller, Sep 18 2011

Array[Boole[And[PrimeQ@ #, PrimeQ[(# - 1)/2]]] &, 105, 0] (* Michael De Vlieger, Dec 16 2017 *)

a(n) = isprime(n) && isprime(floor((n-1)/2)) \\ Iain Fox, Dec 17 2017

a(n) = if n and also (n-1)/2 is prime then 1 else 0;
a(A005385(n)) = 1; a(A156657(n)) = 0; a(A059456(n)) = 0.
a(n) = A010051(n)*A010051((n-1)/2).
A156875(n) = Sum_{k=1..n} a(k). - Reinhard Zumkeller, Feb 18 2009
a(n) = 1 iff A292936(n) > 1. - Antti Karttunen, Dec 15 2017

A156874 Number of Sophie Germain primes <= n.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 1

Reinhard Zumkeller, Feb 18 2009

nonn

a(n) = Sum_{k=1..n} A156660(k).
a(n) = A156875(2*n+1).
Hardy-Littlewood conjecture: a(n) ~ 2*C2*n/(log(n))^2, where C2=0.6601618158... is the twin prime constant (see A005597).
The truth of the above conjecture would imply that there exists an infinity of Sophie Germain primes (which is also conjectured).
a(n) ~ 2*C2*n/(log(n))^2 is also conjectured by Hardy-Littlewood for the number of twin primes <= n.

			a(120) = #{2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113} = 11.

R. Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Sophie Germain prime
Wikipedia, Sophie Germain prime

A156875, A156876, A156877, A156878, A000720.
Cf. A005384 Sophie Germain primes p: 2p+1 is also prime.
Cf. A092816.

Accumulate[Table[Boole[PrimeQ[n]&&PrimeQ[2n+1]],{n,1,200}]] (* Enrique Pérez Herrero, Apr 26 2012 *)
Accumulate[Table[If[AllTrue[{n,2n+1},PrimeQ],1,0],{n,200}]]

a(10^n)= A092816(n). - Enrique Pérez Herrero, Apr 26 2012

Edited and commented by Daniel Forgues, Jul 31 2009

A156876 Number of primes <= n that are safe primes or Sophie Germain primes.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12

Offset: 1

Reinhard Zumkeller, Feb 18 2009

nonn

			a(120) = #{2,3,5,7,11,23,29,41,47,53,59,83,89,107,113} = 15.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A156658, A005384, A005385, A000720.

Cf. A156874, A156875, A156877, A156878.

Accumulate[Table[If[AllTrue[{n,2n+1},PrimeQ]||AllTrue[{n,(n-1)/2}, PrimeQ],1,0],{n,100}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 24 2019 *)

a(n) = my(nb=0); forprime(p=2, n, if (isprime(2*p+1) || isprime((p-1)/2), nb++)); nb; \\ Michel Marcus, Nov 06 2022

a(n) = A156874(n)+A156875(n)-A156877(n) = A000720(n)-A156878(n).

A156877 Number of primes <= n that are safe primes and also Sophie Germain primes.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 1

Reinhard Zumkeller, Feb 18 2009

nonn

			a(120) = #{5, 11, 23, 83} = 4.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A059455, A005384, A005385, A000720.

Cf. A156874, A156875, A156876, A156659, A156660.

a(n) = A156874(n) + A156875(n) - A156876(n).

a(n) = Sum_{k=1..n} A156659(k)*A156660(k).

A174167 Number of safe primes between squares of consecutive primes.

Original entry on oeis.org

2, 2, 1, 3, 1, 3, 2, 4, 4, 2, 7, 4, 1, 6, 3, 10, 1, 10, 6, 2, 10, 5, 12, 9, 9, 4, 6, 3, 9, 26, 6, 10, 5, 18, 4, 17, 11, 10, 17, 13, 3, 23, 3, 9, 6, 36, 32, 8, 6, 9, 15, 10, 22, 19, 18, 15, 7, 22, 15, 9, 31, 43, 13, 6, 14, 47, 25, 35, 10, 10, 21, 32, 23, 18, 9, 27, 34, 18, 32, 46, 3, 38, 12, 20

Offset: 1

Jaspal Singh Cheema, Mar 10 2010

nonn

If you graph n vs a(n), interesting patterns begin to emerge. As you go farther along the n-axis, greater are the number of Safe Primes, on average, within each interval obtained. The smallest count of 1 occurs 4 times (terms: 3rd, 5th, 13th, and 17th) in the sequence above. I suspect the number of Safe Primes within each interval will never be zero. If one could prove this, then it would imply that Safe Primes are infinite. Can you prove it?

			Take any pair of consecutive primes. Let us say the very first one (2,3). Square both terms to obtain an interval (4,9). Within this interval, there are two Safe Primes, namely 5 and 7. Hence the very first term of the sequence above is 2. Similarly, the next term, 2, refers to the two Safe Primes between squares of (3,5), or the interval (9,25), which are 11 and 23.

Jaspal Singh Cheema, Table of n, a(n) for n = 1..10177
Rick Aster, Prime number sieve, SAS prime sieve program
Wikipedia, Safe prime

Cf. A005385, A156875.

Edited by D. S. McNeil, Nov 17 2010

cp's OEIS Frontend

A005385 Safe primes p: (p-1)/2 is also prime.

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A156659 Characteristic function of safe primes.

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A156874 Number of Sophie Germain primes <= n.

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A156876 Number of primes <= n that are safe primes or Sophie Germain primes.

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A156877 Number of primes <= n that are safe primes and also Sophie Germain primes.

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A174167 Number of safe primes between squares of consecutive primes.

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A188719 Number of safe primes < 10^n.

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