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.

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

Author

Hugo Pfoertner, Aug 26 2021

Keywords

• nonn

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

• 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.

Crossrefs

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

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

Programs

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