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A333587 a(n) is the least prime p1 starting an n-tuple of consecutive primes p1, ..., pn of minimal span pn - p1, with first gap p2 - p1 = 4, such that the difference of the occurrence count of these n-tuples and the prediction by the first Hardy-Littlewood conjecture has its first sign change.

%I A333587 #8 Mar 30 2020 21:00:46
%S A333587 5206837,337867,827929093,216646267,251331775687
%N A333587 a(n) is the least prime p1 starting an n-tuple of consecutive primes p1, ..., pn of minimal span pn - p1, with first gap p2 - p1 = 4, such that the difference of the occurrence count of these n-tuples and the prediction by the first Hardy-Littlewood conjecture has its first sign change.
%C A333587 See A333586 for more information and references.
%C A333587 a(2) is the Skewes number for the so-called cousin primes.
%C A333587 The minimal span s(n) = pn - p1 of the n-tuples with an initial gap of 4 is s(2) = 4, s(3) = 6, s(4) = 10, s(5) = 12, s(6) = 16.
%H A333587 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cousin_prime">Cousin prime</a>
%H A333587 Wikipedia, <a href="https://en.wikipedia.org/wiki/Skewes%27s_number">Skewes's number</a>
%o A333587 (PARI) \\ Computes a(3)
%o A333587 Li(x,n)=intnum(t=2,n,1/log(t)^x);
%o A333587 C3=0.635166354604271207206696591272522417342*(9/2); \\ A065418
%o A333587 p1=3;p2=5;n=0;forprime(p=7,10^9,if(p-p1==6&&p-p2==2,n++;d=n-C3*Li(3,p2);if(d>=0,print(p1," ",n,">",C3*Li(3,p));break));p1=p2;p2=p)
%Y A333587 Cf. A023200, A333586.
%K A333587 nonn,hard,more
%O A333587 2,1
%A A333587 _Hugo Pfoertner_, Mar 30 2020