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

1369391, 87613571, 1172531, 21432401, 204540143441, 7572964186421

Offset

2,1

Comments

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.

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.

a(2) is the Skewes number for twin primes, first computed by Wolf (2011).

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.

Links

Table of n, a(n) for n=2..7.

Hugo Pfoertner, Illustration of growth of number of 7-tuples, (2020).

László Tóth, On The Asymptotic Density Of Prime k-tuples and a Conjecture of Hardy and Littlewood, CMST 25(3), 2019, 143-148.

Wikipedia, Skewes's number

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.

Example

For n=6 two types of prime 6-tuples with first gap = 2 starting at p exist:
[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.

Prog

(PARI) Li(x, n)=intnum(t=2, n, 1/log(t)^x);
\\ a(4)
C4=0.307494878758327093123354486071076853*(27/2); \\ A065419
\\ Start at 5 to exclude "fake" 4-tuple 3, 5, 7, 11
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);
\\ a(5)
C5=(15^4/2^11)*0.409874885088236474478781212337955277896358; \\ A269843
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);

Crossrefs

Cf. A005597, A065418, A065419, A269843, A269846, A271742, A333587.

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.

Sequence in context: A250502 A210439 A347278 * A348053 A332493 A234183

Adjacent sequences: A333583 A333584 A333585 * A333587 A333588 A333589

Keyword

nonn,hard,more

Author

Hugo Pfoertner, Mar 30 2020

Extensions

Changed title and clarified definition by Hugo Pfoertner, May 11 2020

Status

approved