login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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 amount 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 amount 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

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.

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-tupel 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

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

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

Sequence in context: A222155 A250502 A210439 * A333586 A348053 A332493

Adjacent sequences:  A347275 A347276 A347277 * A347279 A347280 A347281

KEYWORD

nonn

AUTHOR

Hugo Pfoertner, Aug 26 2021

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 28 22:15 EST 2021. Contains 349415 sequences. (Running on oeis4.)