

A347278


First member p(m) of the mth twin prime pair such that d(m) > 0 and d(m1) < 0, with d(k) = k/Integral_{x=2..p(k)} 1/log(x)^2 dx  C, C = 2*A005597 = A114907.


4



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

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 HardyLittlewood 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 HardyLittlewood 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 HL prediction.


LINKS

Wikipedia, Twin prime, First HardyLittlewood conjecture.


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, (1distres(w, p)/p)/(11/p)^n) * prodeulerrat((1n/p)/(11/p)^n, 1, nextprime(plim+1))}; \\ ktuple 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/LCHL); 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

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


KEYWORD

nonn


AUTHOR



STATUS

approved



