A333587 a(n) is the least prime p1 starting an n-tuple of consecutive primes p1, ..., pn of minimal span pn - p1, with first gap p2 - p1 = 4, such that the difference of the occurrence count of these n-tuples and the prediction by the first Hardy-Littlewood conjecture has its first sign change.

5206837, 337867, 827929093, 216646267, 251331775687

Offset: 2

Author

Hugo Pfoertner, Mar 30 2020

Keywords

• nonn
• hard
• more

Comments

See A333586 for more information and references.

a(2) is the Skewes number for the so-called cousin primes.

The minimal span s(n) = pn - p1 of the n-tuples with an initial gap of 4 is s(2) = 4, s(3) = 6, s(4) = 10, s(5) = 12, s(6) = 16.

Links

• Wikipedia, Cousin prime
• Wikipedia, Skewes's number

Crossrefs

Cf. A023200, A333586.

Programs

PARI

\\ Computes a(3)
Li(x,n)=intnum(t=2,n,1/log(t)^x);
C3=0.635166354604271207206696591272522417342*(9/2); \\ A065418
p1=3;p2=5;n=0;forprime(p=7,10^9,if(p-p1==6&&p-p2==2,n++;d=n-C3*Li(3,p2);if(d>=0,print(p1," ",n,">",C3*Li(3,p));break));p1=p2;p2=p)