A210439 - OEIS

%I #63 Jun 26 2022 09:16:26

%S 1369391,337867,1172531,21432401,251331775687,7572964186421,

%T 1203255673037261

%N The minimal Skewes number for prime n-tuplets.

%C More formally: the least prime in the prime n-tuplet at which for the first time pi_n(p) > C_n*Li_n(p). Here pi_n(p) is the n-tuplet counting function; C_n is the Hardy-Littlewood constant, and Li_n(x) is the integral from 2 to x of (1/(log t)^n) dt.

%C If, for a given n, there is more than one type of n-tuplets, then a(n) is determined by the n-tuplet type for which the first sign change of pi_n - C_n*Li_n occurs earlier than for the other type(s).

%C For the special case n=1, the term a(1) is the Skewes number, i.e., the first prime p for which pi(p) > Li(p). The term a(1) is not included in the sequence because it is not precisely known.

%H Tony Forbes and Norman Luhn, <a href="https://pzktupel.de/ktpatt_hl.php">Patterns of prime k-tuplets & the Hardy-Littlewood constants</a>.

%H Tony Forbes and Norman Luhn, <a href="http://www.pzktupel.de/ktuplets">Prime k-tuplets</a>.

%H Alexei Kourbatov, <a href="/A210439/a210439_1.txt">PARI code for computing a(6)</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="/A210439/a210439.png">Illustration of 8-tuplet counts determining a(8)</a>.

%H László Tóth, <a href="https://arxiv.org/abs/1910.02636">On The Asymptotic Density Of Prime k-tuples and a Conjecture of Hardy and Littlewood</a>, arXiv:1910.02636 [math.NT], 2019.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/k-TupleConjecture.html">k-Tuple Conjecture</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeConstellation.html">Prime Constellation</a>

%H Marek Wolf, <a href="http://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], 2011.

%e Initially, for twin primes we have pi_2(p) < C_2 Li_2(p). The inequality is reversed for the first time for the 10744th pair of twin primes (1369391,1369393), therefore a(2) = 1369391.

%e Similarly, for prime triples (p,p+4,p+6), pi_3(p) < C_3 Li_3(p) until the 652nd triple (337867,337871,337873) where the inequality is reversed for the first time. Thus a(3)=337867. (The reversal for the other type of triples (p,p+2,p+6) occurs much later, so triples (p,p+2,p+6) do not contribute a term to this sequence.)

%e From _Hugo Pfoertner_, 2021 Aug 26, 2021 Oct 24: (Start)

%e a(8) corresponds to the 134292-th 8-tuple of the form p + [0, 2, 6, 8, 12, 18, 20, 26], found using a program provided by _Norman Luhn_. This type of 8-tuple is the one that leads to the earliest crossing of the corresponding comparison value (see linked illustration), while the other two possible configurations (enumerated in A022012 and A022013 or in A346997 and A346998) are still far from crossing their respective applicable comparison values. The other two possible 8-tuples, which lead to the crossing that occurs later, determine the terms A332493(8) and A348053(8), dependent on the criterion applied to decide what is "later". (End)

%o (PARI) See A.Kourbatov link.

%Y Cf. A052435 (round(li(n)-pi(n)), where li is the logarithmic integral and pi(x) is the prime counting function).

%Y Cf. A332493, A333586, A333587, A348053.

%Y Cf A022011, A022012, A022013, A346996, A346997, A346998 (related to 8-tuplets).

%K nonn,hard,more

%O 2,1

%A _Alexei Kourbatov_, Jan 20 2013

%E a(7) from _Hugo Pfoertner_, May 09 2020

%E a(8) from _Hugo Pfoertner_, Aug 26 2021