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%I #77 Oct 24 2021 08:42:10
%S 1369391,87613571,1172531,216646267,251331775687,214159878489239,
%T 750247439134737983
%N The minimal Skewes number for prime n-tuplets, choosing the n-tuplet with latest occurrence of the first sign change relative to the Hardy-Littlewood prediction when more than one type of n-tuplets exists (A083409(n)>1) for the given n.
%C a(n) >= A210439(n). Equals A210439(n) at n=2,4,6, i.e., at those n for which there is only one type of prime n-tuplets (admissible prime n-tuples of minimal span). The corresponding minimal span (diameter) is given by A008407(n).
%C See A210439 for more information, references and links.
%C From _Hugo Pfoertner_, Oct 21 2021: (Start)
%C There are two options for choosing a(8):
%C Either one interprets "latest occurrence" as the largest number of 8-tuplets before the Hardy-Littlewood (H-L) prediction is exceeded, or one selects the larger value of the first 8-tuplet term causing the first crossing.
%C In the first case, 40634356 8-tuplets of the type p + [0, 2, 6, 12, 14, 20, 24, 26] are required before the H-L prediction is exceeded with an 8-tuplet 523250002674163757 + [0, 2, 6, ...].
%C In the second case, 20316822 8-tuplets of type p + [0, 6, 8, 14, 18, 20, 24, 26] are needed to reach the first crossing of the H-L prediction. The corresponding 8-tuplet has 750247439134737983 as first term.
%C The interchanging is a consequence of the different H-L constants for the two tuplet types, 475.36521.. vs. 178.261954.., which have a ratio of 8/3 to one another.
%C Since the H-L constant for the "earliest occurrence" A210439(8) is 178.26.., this speaks in favor of a choice from the two possibilities, which uses the same H-L constant, i.e., the occurrence with the larger tuplet start and not the occurrence with the larger number of required tuplets, for which a separate sequence A348053 is created. (End)
%H Tony Forbes and Norman Luhn, <a href="http://www.pzktupel.de/ktuplets">Prime k-tuplets</a>
%H Alexei Kourbatov, <a href="/A332493/a332493.gp.txt">Optimized PARI code for computing a(7)</a>
%H Norman Luhn, <a href="http://www.pzktupel.de/smarchive.html">Database of the smallest prime k-tuplets</a>, compressed files.
%H Hugo Pfoertner, <a href="/A332493/a332493.pdf">Illustration of growth of number of 7-tuples</a>, (2020).
%H Hugo Pfoertner, <a href="/A332493/a332493_1.pdf">Comparison of number of octuplets needed to achieve the H-L prediction</a>, (2021).
%e Denote by pi_n(x) the n-tuplet counting function, C_n the corresponding Hardy-Littlewood constant, and Li_n(x) the integral from 2 to x of (1/(log t)^n) dt.
%e For 7-tuples with pattern (0 2 8 12 14 18 20) we have the Skewes number p=214159878489239; this is the initial prime p in the 7-tuple where for the first time we have pi_7(p) > C_7 Li_7(p). For the other dense pattern (0 2 6 8 12 18 20), the first sign change of pi_7(x) - C_7 Li_7(x) occurs earlier, at 7572964186421. Therefore we have a(7)=214159878489239, while A210439(7)=7572964186421.
%o (PARI) See A. Kourbatov link.
%o (PARI) \\ The first result is A210439(5), the 2nd is a(5)
%o Li(x, n)=intnum(t=2, n, 1/log(t)^x);
%o G5=(15^4/2^11)*0.409874885088236474478781212337955277896358; \\ A269843
%o n1=0;n2=0;n1found=0;n2found=0;p1=5;p2=7;p3=11;p4=13;
%o forprime(p5=17,10^12,if(p5-p1==12,my(L=Li(5,p1));if(p2-p1==2,n1++;if(!n1found&&n1/L>G5,print(p1," ",p2," ",n1," ",n1/L);n1found=1),n2++;if(!n2found&&n2/L>G5,print(p1," ",p2," ",n2," ",n2/L);n2found=1)));if(n1found&&n2found,break);p1=p2;p2=p3;p3=p4;p4=p5) \\ _Hugo Pfoertner_, May 12 2020
%o \\ Code for a(7), similar to A. Kourbatov's code but much shorter.
%o \\ Run time approx. 2 days, prints every 1000th 7-tuple
%o G7=(35^6/(3*2^22))*0.36943751038649868932319074987675; \\ A271742
%o s=[0,2,8,12,14,18,20];
%o r=[809, 2069, 2909, 5639, 6689, 7529, 7739, 8999, 10259, 12149, 12359, 14459, 14879, 15929, 17189, 19289, 20549, 21389, 23909, 24119, 26009, 27479, 28529, 28739];
%o forstep(p0=0,10^15,30030,for(j=1,24,my(p1=p0+r[j],isp=1,L);for(k=1,7,my(p=p1+s[k]);if(!ispseudoprime(p),isp=0;break));if(isp,L=Li(7,p1);n++;if(n%1000==0||n/L>G7,print(p1," ",p1+s[#s]," ",n/L," ",n));if(n/L>G7,break(2))))) \\ _Hugo Pfoertner_, May 16 2020
%Y Cf. A008407, A083409, A210439, A333586, A333587, A348053.
%K nonn,hard,more
%O 2,1
%A _Alexei Kourbatov_ and _Hugo Pfoertner_, May 11 2020
%E a(8) from _Norman Luhn_ and _Hugo Pfoertner_, Oct 21 2021
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