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 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

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