

A333586


Skewes numbers for prime ntuples p1, p2, ..., pn, with p2  p1 = 2.


5




OFFSET

2,1


COMMENTS

a(n) is the least prime p1 starting an ntuple 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 ntuples and the prediction by the first HardyLittlewood conjecture has its first sign change. When more than one such tuple exists, the ntuple with the lexicographically earliest sequence of gaps is chosen.
These primes are called Skewes's (or Skewes) numbers for prime ktuples 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 ntuples 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 7tuples, (2020).
László Tóth, On The Asymptotic Density Of Prime ktuples and a Conjecture of Hardy and Littlewood, CMST 25(3), 2019, 143148.
Wikipedia, Skewes's number
Wikipedia, Twin prime, First HardyLittlewood 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 6tuples 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 HardyLittlewood prediction for the number of such 6tuples 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 15049th 6tuple 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" 4tuple 3, 5, 7, 11
p1=5; p2=7; p3=11; n=0; forprime(p=13, 10^9, if(pp1==8&&pp2==6, n++; d=nC4*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(pp1==12&&pp2==10, n++; d=nC5*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 ntuplets with minimal span, irrespective of the first gap, is A210439, and its variant A332493.
Sequence in context: A222155 A250502 A210439 * A332493 A234183 A235855
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



