Search: a005597 -id:a005597
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A114907
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Decimal expansion of twice the twin primes constant defined in A005597.
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+20
8
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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
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OFFSET
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1,2
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LINKS
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FORMULA
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Equals 2*A005597 (in the sense of the corresponding decimal numbers).
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EXAMPLE
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1.320323631693739147855624220...
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PROG
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(PARI) 2 * prodeulerrat(1-1/(p-1)^2, 1, 3) \\ Amiram Eldar, Mar 16 2021
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A347278
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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.
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+20
4
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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Wikipedia, Twin prime, First Hardy-Littlewood conjecture.
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PROG
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(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
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CROSSREFS
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a(1) = A210439(2) (Skewes number for twin primes).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A347279
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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.
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+20
3
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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
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OFFSET
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1,1
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COMMENTS
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LINKS
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FORMULA
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PROG
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(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]))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A065426
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Incrementally larger terms in the continued fraction (A065645) for the twin prime constant (A005597).
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+20
0
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OFFSET
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1,3
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LINKS
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MATHEMATICA
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(* 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} ]
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PROG
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(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]))};
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CROSSREFS
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KEYWORD
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nonn,hard,more
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AUTHOR
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STATUS
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approved
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A238734
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Log of twice the twin prime constant, C_2, log(2*A005597).
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+20
0
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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
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OFFSET
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0,1
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COMMENTS
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The value occurs as term in equation (15) in the Wolf paper. - Ralf Stephan, Mar 28 2014
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LINKS
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FORMULA
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EXAMPLE
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0.2778768820732319619323108667032534203602062941473682988245270533677164980...
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MATHEMATICA
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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}];
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A005385
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Safe primes p: (p-1)/2 is also prime.
(Formerly M3761)
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+10
246
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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
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1,1
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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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].
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FORMULA
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MAPLE
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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
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MATHEMATICA
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Select[Prime[Range[1000]], PrimeQ[(#-1)/2]&] (* Zak Seidov, Jan 26 2011 *)
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PROG
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(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
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CROSSREFS
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Except for the initial term, this is identical to A079148.
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Feb 15 2001
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STATUS
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approved
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A065421
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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.
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+10
26
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OFFSET
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1,2
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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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.
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FORMULA
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(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)
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EXAMPLE
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(1/3 + 1/5) + (1/5 + 1/7) + (1/11 + 1/13) + ... = 1.902160583209 +- 0.000000000781 [Nicely]
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CROSSREFS
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Cf. A005597 (twin prime constant Product_{ p prime >= 3 } (1-1/(p-1)^2)).
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KEYWORD
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AUTHOR
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EXTENSIONS
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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
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STATUS
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approved
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A156874
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Number of Sophie Germain primes <= n.
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+10
12
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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
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OFFSET
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1,3
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COMMENTS
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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.
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LINKS
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FORMULA
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EXAMPLE
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a(120) = #{2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113} = 11.
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MATHEMATICA
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CROSSREFS
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Cf. A005384 Sophie Germain primes p: 2p+1 is also prime.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A092816
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Number of Sophie Germain primes less than 10^n.
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+10
9
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3, 10, 37, 190, 1171, 7746, 56032, 423140, 3308859, 26569515, 218116524, 1822848478, 15462601989, 132822315652
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OFFSET
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1,1
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COMMENTS
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Hardy-Littlewood conjecture: Number of Sophie Germain primes less than 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 are an infinite number of Sophie Germain primes (which is also conjectured). - Robert G. Wilson v, Jan 31 2013
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REFERENCES
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P. Ribenboim, The Little Book of Big Primes, Springer-Verlag, New York, 1991, p. 228.
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LINKS
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FORMULA
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For 1 < n < 15, a(n) ~ e * (pi(2*10^n) - pi(10^n)) / (5*n - 5) where e is Napier's constant, see A001113 (we use n > 1 to avoid division by zero; whether the formula holds for any n > 14 is unknown). - Sergey Pavlov, Apr 07 2021 [This formula fails under the Hardy-Littlewood conjecture; the leading constant is wrong. - Charles R Greathouse IV, Aug 03 2023]
For any n, a(n) = qcc(x) - (10^n - pi(10^n) - pi(2 * 10^n + 1) + 1) where qcc(x) is the number of "common composite numbers" c <= 10^n such that both c and c' = 2*c + 1 are composite (trivial). - Sergey Pavlov, Apr 08 2021
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EXAMPLE
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The Sophie Germain primes up to 10 are 2 (since 5 is prime), 3 (since 7 is prime), and 5 (since 11 is prime), so a(1) = 3.
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A274121
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The gap prime(n+1) - prime(n) occurs for the a(n)-th time.
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+10
8
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1, 1, 2, 1, 3, 2, 4, 3, 1, 5, 2, 4, 6, 5, 3, 4, 7, 5, 6, 8, 6, 7, 7, 1, 8, 9, 9, 10, 10, 1, 11, 8, 11, 1, 12, 9, 10, 12, 11, 12, 13, 2, 14, 13, 15, 1, 2, 14, 16, 15, 13, 17, 3, 14, 15, 16, 18, 17, 16, 19, 4, 2, 17, 20, 18, 3, 18, 5, 21, 19, 19, 2, 20, 21, 20, 22, 3, 21, 4, 6, 22, 7, 23, 23, 22, 24, 5, 23, 24, 24, 3, 6
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OFFSET
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1,3
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COMMENTS
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Terms of this sequence grow without bound; any even number occurs in this sequence. Zhang proved that there are infinitely many primes 4680 apart from each other (see link "Bounded gaps between primes").
For a conjectured count of gap n below x, see link Polignac's conjecture.
Polignac's conjecture states that "For any positive even number n, there are infinitely many prime gaps of size n.". By this conjecture, every positive apppears infinitely many times in this sequence (see link "Polignac's conjecture").
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LINKS
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FORMULA
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EXAMPLE
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(p, g) denotes a prime p and the gap up to the next prime. So p + g is the next prime after p. These pairs start (2, 1), (3, 2), (5, 2), (7, 4), (11, 2). From here we see that:
- the gap after the first prime, 1 occurs for the first time, so a(1) = 1.
- the gap after the second prime, 2, occurs for the first time, so a(2) = 1.
- the gap after the third prime, 2, occurs for the second time, so a(3) = 2.
- the gap after the fourth prime, 4, occurs for the first time, so a(4) = 1.
- the gap after the fifth prime, 2, occurs for the third time, so a(5) = 3.
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PROG
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(PARI) \\ See link by name "PARI program" for an extended version with comments.
upto(n) = {my(gapcount=List(), freqgap = List([1])); n = max(n, 3); forprime(i=3, n,
g = nextprime(i+1) - i; for(i=#gapcount+1, g\2, listput(gapcount, 0)); gapcount[g\2]++; listput(freqgap, gapcount[g\2])); freqgap} \\ David A. Corneth, Jun 28 2016
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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