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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. 3
1369391, 87613571, 1172531, 216646267, 251331775687, 214159878489239 (list; graph; refs; listen; history; text; internal format)
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.

LINKS

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

Tony Forbes, Prime k-tuplets

Alexei Kourbatov, Optimized PARI code for computing a(7)

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

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

Cf. A008407, A083409, A210439, A333586, A333587.

Sequence in context: A250502 A210439 A333586 * A234183 A235855 A191820

Adjacent sequences:  A332490 A332491 A332492 * A332494 A332495 A332496

KEYWORD

nonn,hard,more

AUTHOR

Alexei Kourbatov and Hugo Pfoertner, May 11 2020

STATUS

approved

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Last modified July 30 22:16 EDT 2021. Contains 346365 sequences. (Running on oeis4.)