

A210439


The minimal Skewes number for prime ntuplets.


8




OFFSET

2,1


COMMENTS

More formally: the least prime in the prime ntuplet at which for the first time pi_n(p) > C_n*Li_n(p). Here pi_n(p) is the ntuplet counting function; C_n is the HardyLittlewood constant, and Li_n(x) is the integral from 2 to x of (1/(log t)^n) dt.
If, for a given n, there is more than one type of ntuplets, then a(n) is determined by the ntuplet type for which the first sign change of pi_n  C_n*Li_n occurs earlier than for the other type(s).
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.


LINKS



EXAMPLE

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.
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.)
a(8) corresponds to the 134292th 8tuple of the form p + [0, 2, 6, 8, 12, 18, 20, 26], found using a program provided by Norman Luhn. This type of 8tuple 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 8tuples, 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)


PROG

(PARI) See A.Kourbatov link.


CROSSREFS

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


KEYWORD

nonn,hard,more


AUTHOR



EXTENSIONS



STATUS

approved



