A363500 Numbers k between twin primes p, q where k+p and k+q are also twin primes, and k*p and k*q are between twin primes.

6, 109505970, 1519435260, 22606027290, 25980888360, 33995114580, 42029719620, 45284475810, 56527358160, 63402770550, 73924546080, 82625597670, 121883654550, 150444654360, 192416460810, 210205659510, 258719413680, 270709718160, 284455564050, 309050171430

Offset: 1

Author

Bryce Case, Jr. and Antonio Gimenez, Jun 05 2023

Keywords

• nonn

Comments

Larger twin primes are found on either side of 6x, so my highly-unoptimized code simply keeps adding 6 and performing the requisite primality checks using golang's "ProbablyPrime" function, a combination of Miller-Rabin and Baillie-PSW, accurate up to 2^64. Based on seminal work by fellow OEIS contributor Antonio Gimenez.

To generate, k = 6x.

p = k-1, q = k+1, check the primality of k+p, k+q, then check the primality of ((k*p) +/- 1) and ((k*q) +/- 1).

If k > x+1 and x > 1, then all eight primes are not divisible by x. If k > 8, then k == 0 (mod 210). - Jason Yuen, Jun 02 2024

Links

• Jason Yuen, Table of n, a(n) for n = 1..10000 (first 49 terms from Bryce Case, Jr.)
• Bryce Case, Jr., a363500.go

Crossrefs

Subsequence of A066388.

Cf. A364263.

Programs

Go

// See link.

Formula

a(n) = 210*A364263(n-1) for n > 1. - Hugo Pfoertner, Jun 03 2024