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A333586 Skewes numbers for prime n-tuples p1, p2, ..., pn, with p2 - p1 = 2.

%I A333586 #24 May 12 2020 09:14:18
%S A333586 1369391,87613571,1172531,21432401,204540143441,7572964186421
%N A333586 Skewes numbers for prime n-tuples p1, p2, ..., pn, with p2 - p1 = 2.
%C A333586 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 = 2, 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. When more than one such tuple exists, the n-tuple with the lexicographically earliest sequence of gaps is chosen.
%C A333586 These primes are called Skewes's (or Skewes) numbers for prime k-tuples in analogy to the definition for single primes. See Tóth's article for details.
%C A333586 a(2) is the Skewes number for twin primes, first computed by Wolf (2011).
%C A333586 The minimal span s(n) = pn - p1 of the n-tuples with an initial gap of 2 is s(2) = 2, s(3) = 6, s(4) = 8, s(5) = 12, s(6) = 18, s(7) = 20, s(8) = 26.
%H A333586 Hugo Pfoertner, <a href="/A333586/a333586.pdf">Illustration of growth of number of 7-tuples</a>, (2020).
%H A333586 László Tóth, <a href="http://cmst.eu/wp-content/uploads/files/10.12921_cmst.2019.0000033_TOTH.pdf">On The Asymptotic Density Of Prime k-tuples and a Conjecture of Hardy and Littlewood</a>, CMST 25(3), 2019, 143-148.
%H A333586 Wikipedia, <a href="https://en.wikipedia.org/wiki/Skewes%27s_number">Skewes's number</a>
%H A333586 Wikipedia, <a href="https://en.wikipedia.org/wiki/Twin_prime#First_Hardy%E2%80%93Littlewood_conjecture">Twin prime</a>, First Hardy-Littlewood conjecture.
%H A333586 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.
%e A333586 For n=6 two types of prime 6-tuples with first gap = 2 starting at p exist:
%e A333586 [p, p+2, p+6, p+8, p+12, p+18] and [p, p+2, p+8, p+12, p+14, p+18]. The first one has the lexicographically earlier sequence of gaps and is therefore chosen. The Hardy-Littlewood prediction for the number of such 6-tuples with p <= P is (C_6*15^5/2^13)*Integral_{x=2..P} 1/log(x)^6 dx with C_6 given in A269846. The 15049-th 6-tuple starting with a(6)=204540143441 is the first one for which n/Integral_{x=2..a(6)} 1/log(x)^6 dx = 17.29864469487 exceeds C_6*15^5/2^13 = 17.29861231158.
%o A333586 (PARI) Li(x, n)=intnum(t=2, n, 1/log(t)^x);
%o A333586 \\ a(4)
%o A333586 C4=0.307494878758327093123354486071076853*(27/2); \\ A065419
%o A333586 \\ Start at 5 to exclude "fake" 4-tuple 3, 5, 7, 11
%o A333586 p1=5; p2=7; p3=11; n=0; forprime(p=13, 10^9, if(p-p1==8&&p-p2==6, n++; d=n-C4*Li(4, p3); if(d>=0, print(p1, " ", n, ">", C4*Li(4, p)); break)); p1=p2; p2=p3; p3=p);
%o A333586 \\ a(5)
%o A333586 C5=(15^4/2^11)*0.409874885088236474478781212337955277896358; \\ A269843
%o A333586 p1=3; p2=5; p3=7; p4=11; n=0; forprime(p=13, 10^9, if(p-p1==12&&p-p2==10, n++; d=n-C5*Li(5, p4); if(d>=0, print(p1, " ", n, ">", C5*Li(5, p)); break)); p1=p2; p2=p3; p3=p4; p4=p);
%Y A333586 Cf. A005597, A065418, A065419, A269843, A269846, A271742, A333587.
%Y A333586 The sequence of Skewes numbers always choosing the prime n-tuplets with minimal span, irrespective of the first gap, is A210439, and its variant A332493.
%K A333586 nonn,hard,more
%O A333586 2,1
%A A333586 _Hugo Pfoertner_, Mar 30 2020
%E A333586 Changed title and clarified definition by _Hugo Pfoertner_, May 11 2020