|
| |
|
|
A332493
|
|
The minimal Skewes number for prime n-tuplets, choosing the n-tuplet with latest occurrence of the first sign change relative to the Hardy-Littlewood prediction when more than one type of n-tuplets exists (A083409(n)>1) for the given n.
|
|
8
|
| |
|
|
|
OFFSET
|
2,1
|
|
|
COMMENTS
|
a(n) >= A210439(n). Equals A210439(n) at n=2,4,6, i.e., at those n for which there is only one type of prime n-tuplets (admissible prime n-tuples of minimal span). The corresponding minimal span (diameter) is given by A008407(n).
See A210439 for more information, references and links.
There are two options for choosing a(8):
Either one interprets "latest occurrence" as the largest number of 8-tuplets before the Hardy-Littlewood (H-L) prediction is exceeded, or one selects the larger value of the first 8-tuplet term causing the first crossing.
In the first case, 40634356 8-tuplets of the type p + [0, 2, 6, 12, 14, 20, 24, 26] are required before the H-L prediction is exceeded with an 8-tuplet 523250002674163757 + [0, 2, 6, ...].
In the second case, 20316822 8-tuplets of type p + [0, 6, 8, 14, 18, 20, 24, 26] are needed to reach the first crossing of the H-L prediction. The corresponding 8-tuplet has 750247439134737983 as first term.
The interchanging is a consequence of the different H-L constants for the two tuplet types, 475.36521.. vs. 178.261954.., which have a ratio of 8/3 to one another.
Since the H-L constant for the "earliest occurrence" A210439(8) is 178.26.., this speaks in favor of a choice from the two possibilities, which uses the same H-L constant, i.e., the occurrence with the larger tuplet start and not the occurrence with the larger number of required tuplets, for which a separate sequence A348053 is created. (End)
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
Denote by pi_n(x) the n-tuplet counting function, C_n the corresponding Hardy-Littlewood constant, and Li_n(x) the integral from 2 to x of (1/(log t)^n) dt.
For 7-tuples with pattern (0 2 8 12 14 18 20) we have the Skewes number p=214159878489239; this is the initial prime p in the 7-tuple where for the first time we have pi_7(p) > C_7 Li_7(p). For the other dense pattern (0 2 6 8 12 18 20), the first sign change of pi_7(x) - C_7 Li_7(x) occurs earlier, at 7572964186421. Therefore we have a(7)=214159878489239, while A210439(7)=7572964186421.
|
|
|
PROG
|
(PARI) See A. Kourbatov link.
(PARI) \\ The first result is A210439(5), the 2nd is a(5)
Li(x, n)=intnum(t=2, n, 1/log(t)^x);
G5=(15^4/2^11)*0.409874885088236474478781212337955277896358; \\ A269843
n1=0; n2=0; n1found=0; n2found=0; p1=5; p2=7; p3=11; p4=13;
forprime(p5=17, 10^12, if(p5-p1==12, my(L=Li(5, p1)); if(p2-p1==2, n1++; if(!n1found&&n1/L>G5, print(p1, " ", p2, " ", n1, " ", n1/L); n1found=1), n2++; if(!n2found&&n2/L>G5, print(p1, " ", p2, " ", n2, " ", n2/L); n2found=1))); if(n1found&&n2found, break); p1=p2; p2=p3; p3=p4; p4=p5) \\ Hugo Pfoertner, May 12 2020
\\ Code for a(7), similar to A. Kourbatov's code but much shorter.
\\ Run time approx. 2 days, prints every 1000th 7-tuple
G7=(35^6/(3*2^22))*0.36943751038649868932319074987675; \\ A271742
s=[0, 2, 8, 12, 14, 18, 20];
r=[809, 2069, 2909, 5639, 6689, 7529, 7739, 8999, 10259, 12149, 12359, 14459, 14879, 15929, 17189, 19289, 20549, 21389, 23909, 24119, 26009, 27479, 28529, 28739];
forstep(p0=0, 10^15, 30030, for(j=1, 24, my(p1=p0+r[j], isp=1, L); for(k=1, 7, my(p=p1+s[k]); if(!ispseudoprime(p), isp=0; break)); if(isp, L=Li(7, p1); n++; if(n%1000==0||n/L>G7, print(p1, " ", p1+s[#s], " ", n/L, " ", n)); if(n/L>G7, break(2))))) \\ Hugo Pfoertner, May 16 2020
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,hard,more
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
