A005597 -id:A005597 - cp's OEIS frontend

A114907 Decimal expansion of twice the twin primes constant defined in A005597.

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

Offset: 1

Eric W. Weisstein, Jan 06 2006

			1.320323631693739147855624220...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.1, p. 85.

Alexei Kourbatov, On the distribution of maximal gaps between primes in residue classes, arXiv:1610.03340 [math.NT], 2016-2018.
Eric Weisstein's World of Mathematics, Twin Primes Constant.
Eric Weisstein's World of Mathematics, Prime Constellation.

Cf. A005597.

2 * prodeulerrat(1-1/(p-1)^2, 1, 3) \\ Amiram Eldar, Mar 16 2021

Equals 2*A005597 (in the sense of the corresponding decimal numbers).

Corrected offset and edited by M. F. Hasler, Dec 09 2008

A347278 First member p(m) of the m-th twin prime pair such that d(m) > 0 and d(m-1) < 0, with d(k) = k/Integral_{x=2..p(k)} 1/log(x)^2 dx - C, C = 2*A005597 = A114907.

Original entry on oeis.org

1369391, 1371989, 1378217, 1393937, 1418117, 1426127, 1428767, 1429367, 1430291, 1494509, 1502141, 1502717, 1506611, 1510307, 35278697, 35287001, 35447171, 35468429, 35468861, 35470271, 35595869, 45274121, 45276227, 45304157, 45306827, 45324569, 45336461, 45336917

Offset: 1

Hugo Pfoertner, Aug 26 2021

nonn

The sequence gives the positions, expressed by A001359(m), where the number of twin prime pairs m seen so far first exceeds the number predicted by the first Hardy-Littlewood conjecture after having been less than the predicted number before. A347279 gives the transitions in the opposite direction.

The total number of twin prime pairs up to that with first member x in the intervals a(k) <= x < A347279(k) is above the Hardy-Littlewood prediction. The total number of twin prime pairs up to that with first member x in the intervals A347279(k) <= x < a(k+1) is below the H-L prediction.

Hugo Pfoertner, Table of n, a(n) for n = 1..12135
Wikipedia, Twin prime, First Hardy-Littlewood conjecture.
Marek Wolf, The Skewes number for twin primes: counting sign changes of pi_2(x)-C_2 Li_2(x), arXiv:1107.2809 [math.NT], 14 Jul 2011.

Cf. A001359, A005597, A114907, A152051, A347279.

a(1) = A210439(2) (Skewes number for twin primes).

halicon(h) = {my(w=Set(vecsort(h)), n=#w, wmin=vecmin(w), distres(v,p)=#Set(v%p)); for(k=1,n, w[k]=w[k]-wmin); my(plim=nextprime(vecmax(w))); prodeuler(p=2, plim, (1-distres(w,p)/p)/(1-1/p)^n) * prodeulerrat((1-n/p)/(1-1/p)^n, 1, nextprime(plim+1))}; \\ k-tuple constant
Li(x, n)=intnum(t=2, n, 1/log(t)^x); \\ logarithmic integral
a347278(nterms,CHL)={my(n=1,pprev=1,np=0); forprime(p=5,, if(p%6!=1&&ispseudoprime(p+2), n++; L=Li(2,p); my(x=n/L-CHL); if(x*pprev>0, if(pprev>0,print1(p,", ");np++; if(np>nterms,return)); pprev=-pprev)))};
a347278(10,halicon([0,2])) \\ computing 30 terms takes about 5 minutes

A347279 First member p(m) of the m-th twin prime pair such that d(m) < 0 and d(m-1) > 0, with d(k) = k/Integral_{x=2..p(k)} 1/log(x)^2 dx - C, C = 2*A005597 = A114907.

Original entry on oeis.org

1371911, 1372757, 1393919, 1417991, 1425881, 1428671, 1429247, 1429859, 1430711, 1495379, 1502687, 1503317, 1510217, 35278601, 35280029, 35446781, 35463497, 35468789, 35469779, 35472137, 45225161, 45274751, 45276689, 45306641, 45324551, 45336407, 45336761, 45337517

Offset: 1

Hugo Pfoertner, Aug 26 2021

nonn

See A347278 for more information.

Hugo Pfoertner, Table of n, a(n) for n = 1..12134

Cf. A001359, A005597, A114907, A347278.

\\ see A347278 for auxiliary functions halicon and Li.
a347279(nterms,CHL) = {my(n=2,pprev=1,np=0);
forprime(p=11,, if(p%6!=1&&ispseudoprime(p+2), n++; L=Li(2,p); my(x=n/L-CHL); if(x*pprev>0, if(pprev<0,print1(p,", "); np++; if(np>nterms,return)); pprev=-pprev)))};
a347279(10, halicon([0,2]))

A347278(n) < a(n) < A347278(n+1).

A065426 Incrementally larger terms in the continued fraction (A065645) for the twin prime constant (A005597).

Original entry on oeis.org

0, 1, 16, 18, 21, 405, 1199, 2301, 19965

Offset: 1

Robert G. Wilson v, Nov 15 2001

Gerhard Niklasch, Some number-theoretical constants: 1000-digit values

Cf. A005597 and A065645.

(* tpc copied from Niklasch reference *)
cof = ContinuedFraction[tpc, 969]; a = -1; k = 1; Do[ While[ cof[[k]] <= a, k++ ]; a = cof[[k]]; Print[a], {n, 1, 9} ]

\\ Increasing lprec to 30000 gives no further term beyond 19965.
a065246(lprec) = {localprec(lprec); my (m=-1, T=prodeulerrat(1-1/(p-1)^2, 1, 3), c=contfrac(T)); for (k=1, #c, if (c[k]>m, print(c[k],", "); m=c[k]))};
a065246(1000) \\ Hugo Pfoertner, Aug 01 2023

A238734 Log of twice the twin prime constant, C_2, log(2*A005597).

Original entry on oeis.org

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

Offset: 0

John W. Nicholson, Mar 03 2014

The value occurs as term in equation (15) in the Wolf paper. - Ralf Stephan, Mar 28 2014

			0.2778768820732319619323108667032534203602062941473682988245270533677164980...

Marek Wolf, Nearest-neighbor-spacing distribution of prime numbers and quantum chaos, Phys. Rev. E 89, 022922 (2014); arXiv preprint, arXiv:1212.3841 [math.NT], 2012-2014.

Cf. A005597, A114907.

digits = 113;
s[n_] := (1/n)*N[Sum[MoebiusMu[d]*2^(n/d), {d, Divisors[n]}], digits + 50];
C2 = (175/256)*Product[(Zeta[n]*(1 - 2^(-n))*(1 - 3^(-n))*(1 - 5^(-n))*(1 - 7^(-n)))^(-s[n]), {n, 2, digits + 50}];
RealDigits[Log[2 C2]][[1]][[1 ;; digits]] (* Jean-François Alcover, Feb 16 2019 *)

default(realprecision,1000);
result={175/256*prod(k=2, 500, (zeta(k)*(1-1/2^k)*(1-1/3^k)*(1-1/5^k)*(1-1/7^k))^(-sumdiv(k, d, moebius(d)*2^(k/d))/k))};log(2*result)

log(2 * prodeulerrat(1-1/(p-1)^2, 1, 3)) \\ Amiram Eldar, Mar 16 2021

Equals log(2*A005597).

A378482 Decimal expansion of 1/(8log(2)A005597), where A005597 is the twin prime constant C_2.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Nov 28 2024

			0.27317072236263839747106601431655151479129736936570...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.5.1, p. 111.
Gérald Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge University Press, 1995, p. 53, exercise 5 (in the third edition 2015, p. 59, exercise 57).

Paul T. Bateman, Proof of a conjecture of Grosswald, Duke Mathematical Journal, Vol. 25. No. 1 (1958), pp. 67-72.
Emil Grosswald, The average order of an arithmetic function, Duke Mathematical Journal, Vol. 23, No. 1 (1956), pp. 41-44.

Cf. A002162, A005597, A069205.

1/(8*log(2)*prodeulerrat(1-1/(p-1)^2, 1, 3)) \\ Amiram Eldar, Nov 29 2024

Equals lim_{n->oo} (1/(n*log(n)^2)) * A069205(n). - Amiram Eldar, Feb 15 2025

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

Original entry on oeis.org

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

A065421 Decimal expansion of Viggo Brun's constant B, also known as the twin primes constant B_2: Sum (1/p + 1/q) as (p,q) runs through the twin primes.

Original entry on oeis.org

1, 9, 0, 2, 1, 6, 0, 5, 8

Offset: 1

Robert G. Wilson v, Sep 08 2000

The calculation of Brun's constant is "based on heuristic considerations about the distribution of twin primes" (Ribenboim, 1989).
Another constant related to the twin primes is the twin primes constant C_2 (sometimes also denoted PI_2) A005597 defined in connection with the Hardy-Littlewood conjecture concerning the distribution pi_2(x) of the twin primes.
Comment from Hans Havermann, Aug 06 2018: "I don't think the last three (or possibly even four) OEIS terms [he is referring to the sequence at that date - it has changed since then] are necessarily warranted. P. Sebah (see link below) (http://numbers.computation.free.fr/Constants/Primes/twin.html) gives 1.902160583104... as the value for primes to 10^16 followed by a suggestion that the (final) value 'should be around 1.902160583...'" - added by N. J. A. Sloane, Aug 06 2018

			(1/3 + 1/5) + (1/5 + 1/7) + (1/11 + 1/13) + ... = 1.902160583209 +- 0.000000000781 [Nicely]

R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 14.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 133-135.
Paulo Ribenboim, The Book of Prime Number Records, 2nd. ed., Springer-Verlag, New York, 1989, p. 201.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 193.

V. Brun, La série 1/5 + 1/7 + 1/11 + 1/13 + 1/17 + 1/19 + 1/29 + 1/31 + 1/41 + 1/43 + 1/59 + 1/61 + ... où les dénominateurs sont "nombres premiers jumeaux" est convergente ou finie, Bull Sci. Math. 43 (1919), 100-104 and 124-128.
C. K. Caldwell, The Prime Glossary, Brun's constant
Sebastian M. Cioabă and Werner Linde, A Bridge to Advanced Mathematics: from Natural to Complex Numbers, Amer. Math. Soc. (2023) Vol. 58, see page 334.
Steven R. Finch, Brun's Constant [Broken link]
Steven R. Finch, Brun's Constant [From the Wayback machine]
Thomas R. Nicely, Enumeration to 10^14 of the twin primes and Brun's constant, Virginia Journal of Science, 46:3 (Fall, 1995), 195-204.
Thomas R. Nicely, Enumeration to 10^14 of the twin primes and Brun's constant [Local copy, pdf only]
Thomas R. Nicely, Prime Constellations Research Project
P. Sebah, Numbers, constants and computation
D. Shanks and J. W. Wrench, Brun's constant, Math. Comp. 28 (1974) 293-299; 28 (1974) 1183; Math. Rev. 50 #4510.
H. Tronnolone, A tale of two primes, COLAUMS Space, #3, 2013.
Wikipedia, Brun's constant

Cf. A005597 (twin prime constant Product_{ p prime >= 3 } (1-1/(p-1)^2)).

Cf. A077800 (twin primes).

Equals Sum_{n>=1} 1/A077800(n).
From Dimitris Valianatos, Dec 21 2013: (Start)
(1/5) + Sum_{n>=1, excluding twin primes 3,5,7,11,13,...} mu(n)/n =
(1/5) + 1 - 1/2 + 1/6 + 1/10 + 1/14 + 1/15 + 1/21 + 1/22 - 1/23 + 1/26 - 1/30 + 1/33 + 1/34 + 1/35 - 1/37 + 1/38 + 1/39 - 1/42 ... = 1.902160583... (End)

Corrected by N. J. A. Sloane, Nov 16 2001
More terms computed by Pascal Sebah (pascal_sebah(AT)ds-fr.com), Jul 15 2001
Further terms computed by Pascal Sebah (psebah(AT)yahoo.fr), Aug 22 2002
Commented and edited by Daniel Forgues, Jul 28 2009
Commented and reference added by Jonathan Sondow, Nov 26 2010
Unsound terms after a(9) removed by Gord Palameta, Sep 06 2018

A049481 Primes p such that p + 30 is also prime.

Original entry on oeis.org

7, 11, 13, 17, 23, 29, 31, 37, 41, 43, 53, 59, 67, 71, 73, 79, 83, 97, 101, 107, 109, 127, 137, 149, 151, 163, 167, 181, 193, 197, 199, 211, 227, 233, 239, 241, 251, 263, 277, 281, 283, 307, 317, 337, 349, 353, 359, 367, 379, 389, 401, 409, 419, 431, 433, 449

Offset: 1

Labos Elemer

nonn

30 = A002110(3) is the 3rd primorial number.

p and p+30 are not necessarily consecutive primes. Initial segment of A045320 is identical, but 113 is not in this sequence because 113 + 30 = 143 is divisible by 13.

			7 is a term since it is prime and 7 + 30 = 37 is also prime.

Harvey P. Dale, Table of n, a(n) for n = 1..10000
Hugo Pfoertner, Observed ratio n*log(a(n))/pi(a(n)) for n=10^7..5.6*10^9 with a conjectured extrapolation for large n (2024).

Cf. A002110, A045320, A005597.

Cf. A001359, A023201, A049482, A049483, A049484, A049485, A154114.

lst={};Do[p=Prime[n];If[PrimeQ[p+30],AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 04 2009 *)
Select[Prime[Range[100]],PrimeQ[#+30]&] (* Harvey P. Dale, Apr 28 2012 *)

isok(p) = isprime(p) && isprime(p + 30); \\ Amiram Eldar, Mar 15 2025

Assuming Polignac's conjecture and the first Hardy-Littlewood conjecture: Limit_{n->oo} n*log(a(n))/primepi(a(n)) = (16/3)*A005597 = 3.52086... . - Alain Rocchelli, Oct 29 2024

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

cp's OEIS Frontend

A114907 Decimal expansion of twice the twin primes constant defined in A005597.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

PARI

Formula

Extensions

A347278 First member p(m) of the m-th twin prime pair such that d(m) > 0 and d(m-1) < 0, with d(k) = k/Integral_{x=2..p(k)} 1/log(x)^2 dx - C, C = 2*A005597 = A114907.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A347279 First member p(m) of the m-th twin prime pair such that d(m) < 0 and d(m-1) > 0, with d(k) = k/Integral_{x=2..p(k)} 1/log(x)^2 dx - C, C = 2*A005597 = A114907.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A065426 Incrementally larger terms in the continued fraction (A065645) for the twin prime constant (A005597).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A238734 Log of twice the twin prime constant, C_2, log(2*A005597).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A378482 Decimal expansion of 1/(8*log(2)*A005597), where A005597 is the twin prime constant C_2.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

A378482 Decimal expansion of 1/(8log(2)A005597), where A005597 is the twin prime constant C_2.