A347278 - OEIS

%I #19 Jul 13 2022 21:58:31

%S 1369391,1371989,1378217,1393937,1418117,1426127,1428767,1429367,

%T 1430291,1494509,1502141,1502717,1506611,1510307,35278697,35287001,

%U 35447171,35468429,35468861,35470271,35595869,45274121,45276227,45304157,45306827,45324569,45336461,45336917

%N 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.

%C 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.

%C 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.

%H Hugo Pfoertner, <a href="/A347278/b347278.txt">Table of n, a(n) for n = 1..12135</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Twin_prime#First_Hardy%E2%80%93Littlewood_conjecture">Twin prime</a>, First Hardy-Littlewood conjecture.

%H Marek Wolf, <a href="https://arxiv.org/abs/1107.2809">The Skewes number for twin primes: counting sign changes of pi_2(x)-C_2 Li_2(x)</a>, arXiv:1107.2809 [math.NT], 14 Jul 2011.

%o (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

%o Li(x, n)=intnum(t=2, n, 1/log(t)^x); \\ logarithmic integral

%o 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)))};

%o a347278(10,halicon([0,2])) \\ computing 30 terms takes about 5 minutes

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

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

%K nonn

%O 1,1

%A _Hugo Pfoertner_, Aug 26 2021