Displaying 1-10 of 40 results found.
Decimal expansion of twice the twin primes constant defined in A005597.
+20
8
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
REFERENCES
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.1, p. 85.
FORMULA
Equals 2* A005597 (in the sense of the corresponding decimal numbers).
EXAMPLE
1.320323631693739147855624220...
PROG
(PARI) 2 * prodeulerrat(1-1/(p-1)^2, 1, 3) \\ Amiram Eldar, Mar 16 2021
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.
+20
4
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
COMMENTS
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.
LINKS
Wikipedia, Twin prime, First Hardy-Littlewood conjecture.
PROG
(PARI) 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
CROSSREFS
a(1) = A210439(2) (Skewes number for twin primes).
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.
+20
3
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
PROG
(PARI) \\ 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]))
Incrementally larger terms in the continued fraction ( A065645) for the twin prime constant ( A005597).
+20
0
0, 1, 16, 18, 21, 405, 1199, 2301, 19965
MATHEMATICA
(* 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} ]
PROG
(PARI) \\ 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]))};
Log of twice the twin prime constant, C_2, log(2* A005597).
+20
0
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
COMMENTS
The value occurs as term in equation (15) in the Wolf paper. - Ralf Stephan, Mar 28 2014
EXAMPLE
0.2778768820732319619323108667032534203602062941473682988245270533677164980...
MATHEMATICA
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}];
PROG
(PARI)
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)
(PARI) log(2 * prodeulerrat(1-1/(p-1)^2, 1, 3)) \\ Amiram Eldar, Mar 16 2021
Decimal expansion of 1/(8*log(2)* A005597), where A005597 is the twin prime constant C_2.
+20
0
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
REFERENCES
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.5.1, p. 111.
EXAMPLE
0.27317072236263839747106601431655151479129736936570...
PROG
(PARI) 1/(8*log(2)*prodeulerrat(1-1/(p-1)^2, 1, 3)) \\ Amiram Eldar, Nov 29 2024
Safe primes p: (p-1)/2 is also prime.
(Formerly M3761)
+10
250
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
COMMENTS
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 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 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 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].
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) [x|x<-primes(10^3), bigomega(x-1)==2] \\ Altug Alkan, Nov 04 2015 (Haskell)
a005385 n = a005385_list !! (n-1)
a005385_list = filter ((== 1) . a010051 . (`div` 2)) a000040_list
(Magma) [p: p in PrimesUpTo(3000) | IsPrime((p-1) 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((p-1)//2): alst.append(p)
return alst
CROSSREFS
Except for the initial term, this is identical to A079148.
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Feb 15 2001
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.
+10
26
1, 9, 0, 2, 1, 6, 0, 5, 8
COMMENTS
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
REFERENCES
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.
P. Ribenboim, The Book of Prime Number Records, 2nd. ed., Springer-Verlag, New York, 1989, p. 201.
LINKS
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. D. Shanks and J. W. Wrench, Brun's constant, Math. Comp. 28 (1974) 293-299; 28 (1974) 1183; Math. Rev. 50 #4510.
FORMULA
(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)
EXAMPLE
(1/3 + 1/5) + (1/5 + 1/7) + (1/11 + 1/13) + ... = 1.902160583209 +- 0.000000000781 [Nicely]
CROSSREFS
Cf. A005597 (twin prime constant Product_{ p prime >= 3 } (1-1/(p-1)^2)).
EXTENSIONS
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
Both p and p+30 are primes.
+10
13
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
COMMENTS
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.
FORMULA
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
EXAMPLE
Both 7 and 7 + 2*3*5 = 37 are prime.
MATHEMATICA
Select[Prime[Range[100]], PrimeQ[#+30]&] (* Harvey P. Dale, Apr 28 2012 *)
Number of Sophie Germain primes <= n.
+10
12
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
COMMENTS
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.
EXAMPLE
a(120) = #{2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113} = 11.
CROSSREFS
Cf. A005384 Sophie Germain primes p: 2p+1 is also prime.
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