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

2,1

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

Table of n, a(n) for n=2..6.

Wikipedia, Cousin prime

Wikipedia, Skewes's number

Prog

(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)

Crossrefs

Cf. A023200, A333586.

Sequence in context: A257153 A257160 A254742 * A125780 A294988 A183746

Adjacent sequences: A333584 A333585 A333586 * A333588 A333589 A333590

Keyword

nonn,hard,more

Author

Hugo Pfoertner, Mar 30 2020

Status

approved