A266953 Least prime p such that p-2 and 6n-p and either 6n+2-p or 6n+4-p is also prime, or 0 if no such prime exists.

0, 0, 5, 5, 5, 13, 7, 5, 5, 13, 19, 7, 5, 7, 5, 19, 0, 5, 5, 5, 13, 19, 5, 31, 7, 13, 7, 13, 5, 73, 31, 7, 13, 5, 7, 13, 19, 31, 5, 5, 13, 7, 13, 19, 73, 31, 7, 5, 7, 13, 19, 109, 5, 5, 13, 19, 109, 31, 109, 5, 13, 19, 61, 31, 5, 43, 199, 5, 61, 103, 73, 7, 13, 7, 5, 19, 109, 5, 5, 13, 19, 139, 5, 151, 5, 199, 0, 61, 7, 13, 19, 199, 31, 139, 43, 109, 7, 13, 19

Offset: 0

Author

M. F. Hasler, Jan 06 2016

Keywords

• nonn

Comments

If a(n) > 0, then the triple {6n-2, 6n, 6n+2} of consecutive even numbers allows a "simultaneous Goldbach decomposition" using only 4 different primes, 6n-2 = p-2 + 6n-p ; 6n = p + 6n-p ; 6n+2 = p + 6n+2-p = p-2 + 6n+4-p.

See A266952 for the version which does not allow the second decomposition of the last member. See A266948 for a variant which does not require 6n+2-p to be prime.

Up to 10^5, the only indices for which a(n)=0 are {0, 1, 16, 86, 131, 151, 186, 191, 211, 226, 541, 701}. (Only 2 and 67 require the alternative primality of 6n+4-p and have thus A266952(n)=0.) I conjecture that this list is finite, and probably complete. Is it a coincidence that all odd numbers in this list are primes?

Programs

PARI

A266953(n)=my(GP(n, p=2)=forprime(p=p,n+1,isprime(n*2-p)&&return(p))); for(p=1,3*n,isprime(-2+p=GP(3*n, p))+!p&&(!p||isprime(6*n+2-p)||isprime(6*n+4-p))&&return(p))